Ken Roberts

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I think that an understanding of the physics and biomechanics of (land-based) muscle-powered propulsion leads to a single reasonable approach for how to find the optimal "motion technique" for speed for each mode of propulsion (e.g. bicycling, running, walking, skating, cross-country skiing).

I wrote this because much of this website is a detailed explanation of the solution to the problem of finding maximum speed -- actually several solutions for several different situations. But it can be difficult to understand why it's a helpful solution without also understanding some key aspects of the physics and biomechanics of the "problem" -- the problem of speed for muscle-powered propulsion -- and how those key aspects lead to a certain approach for the optimal solution.

Otherwise we end up with arguments over details of the proposed solution that are based on assumptions or approaches which cannot be helpful, based on a careful understanding of the fundamental physics and biomechanics of the problem situation.

I explain here:

  • what this approach is

  • how it fits the basic physics and biomechanics

  • why it's better than alternative approaches

  • what are the practical implications for improving someone's personal motion techniques for speed and training the body for speed.

quick summary of my approach

My approach is to examine many different muscle moves available, and see which ones (together with their corresponding recovery moves) can add some net positive work which increases the average rate of power over the stroke-cycle -- without much hindering the work from other moves or increasing the resistive forces against speed. I'm happy to add non-obvious small moves which exploit clever physics to add small propulsive benefits, as long as they fit in easily with the obvious big moves.

Also to look for ways to change the timing of muscle moves, to increase the propulsive work from a move. Or to overlap moves (or at least shrink some power "dead spots" or "low spots") to reduce the total time of each stroke-cycle. There's some other aspects of my approach, but those are the major ones.

I think my approach is good because: (a) It fits the key characteristics of the physics of the situation; (b) I've found it explains my own propulsive motions and helps me improve my speed with them; (c) It explains the many non-obvious moves I observe in video clips of elite performers in actual muscle-powered race events.  

goals of optimization

There are several possible goals for propulsion:

  • maximum speed (over a given distance or time)

  • feeling good (including long glide)

  • looking good

  • burning lots of calories -- maximum energy over given time

  • minimum energy for given distance and terrain

  • etc.

Any of those goals might be valid for some skater during some hour or day in some place.

Speed -- We're going to focus here on this page on optimal technique for maximum speed. It might be speed in a short sprint. It might be speed for three hours. It might be speed for a short section during a much longer event with other very different sections which require different optimal techniques.

No "right" or "proper" technique

Contrary to what some teachers and coaches seem to assume, there is no reason to think there is one right technique for a given mode of propulsion.

  • Different people might have different goals, which might result in different "best" techniques, because optimized for a different goal.

  • Different people have different capabilities, which might require different techniques, because optimized for different constraints.

  • Different terrain and competitive-strategy situations might require different techniques, because optimized for different terrain constraints or strategic sub-goals.


I'm going to assume that the hilliness and wind and ground-surface resistance are approximately constant for the time period of interest. I recognize that's hardly ever true for an entire publicly-measured / compared human muscle-powered speed performance. But I'm going to assume that this analysis is focusing on some subsection of the performance for which this "constancy" assumption holds approximately.

There's a lot to be said about terrain strategy in how to handle uphills followed by downhill and more uphills, or competitive strategy of when to sprint and what to do while waiting for the sprint. But I'm not dealing with those strategy questions here.

Instead I'm going to assume that the larger performance requirements and strategies are known, and the constraints on "sustainable" force and power from different muscle moves have somehow been derived from that larger context.


[ more to be added ]


other approaches for finding best technique

simple central constraint: VO2max

One alternative is to say: "Why bother with detailed analysis of all those possible moves when in the end what really determines maximum speed is VO2max?"

"VO2max" refers to the maximum amount of oxygen that the central cardio-vascular system (lungs, heart, major arteries) can deliver through the blood to the rest of the body. The idea is that muscular work is aerobic or anaerobic. The aerobic work consumes oxygen immediately. The anaerobic processes perform work without oxygen, but to continue to function they must eventually receive the same amount of oxygen to restore their capacity, to repay the "oxygen debt".

So overall muscular exertion in the body is often measured as percentage of VO2max.

strong + simplistic form of this approach

The "strong + simplistic" form of this approach is to say that "technique is irrelevant", since all that matters is finding some combination of muscle moves which "consume" the percentage of VO2max which is known to be sustainable for the time duration of this given performance.

But surely this doesn't make sense, since some muscle moves consume oxygen but are completely irrelevant to this propulsive motion. And among the muscle moves that make some positive contribution of propulsive work, some have a higher percentage of their exertion being productive for propulsion than others.

more refined form

The more reasonable idea is that we should prioritize the list of available muscle moves according to the ratio of net positive propulsive work to oxygen consumption for each. From this list, start adding the highest-ratio muscle moves to our "technique ensemble" -- until this athlete's VO2max percentage for this time-duration of performance is exhausted. Then stop adding moves -- do not attempt to add all the "many" moves available with positive work (as in my approach summarized above)

My quick response to this:

  • If propulsive performance is truly constrained by central cardio-vascular capacity rather than capacity of some specific peripheral muscles, than I would agree with this refined approach. (But muscle-powered propulsive performance is generally not centrally constrained, except for very short-duration performances.)

  • This approach done properly would still (like mine) require carefully analyzing all the possible muscle moves. Unless it is just assumed that the obvious moves already known are also the ones with the highest ratio of propulsive work.

  • Where does this "percentage of VO2max appropriate for this time-duration" number come from (if it's not exactly 100%) in this "simple" theory of a single central cardio-vascular constraint?

eliminate wasted motion

Another approach taken to explain or advise on points of technique is to look for extra moves which are not propulsive, because they are wasting energy. And because if you eliminate those you're more likely to be on the right track to good technique, and you'll find some other moves to use that energy which will be more productive.

This does not sound like it is intended as a "complete" theory of optimal technique.

So is it a helpful addition to some other theory?

I think, perhaps Yes in two widely differing situations:

  • for very short-duration performances, it's similar to the VO2max approach, if the word "energy" is interpreted to mean the use of oxygen.

  • for very long-duration performances (especially if access to food is limited), it could be useful if the word "energy" is interpreted to mean fuel or food which can be converted into energy.

The second point might have two variations:

  • for ultra-marathons, it might refer to the need to ingest additional food. The objection to this is that most ultra-marathons have feed stations, so it is possible to re-fuel. But then eating takes some time. And there is a limit to the rate at which the body can absorb energy-bearing fuel from food -- and the risk of digestive upset.

  • for performances shorter than two hours, my guess is that the human body can support most of the propulsive work by using stored glycogen as fuel (with the aid of some "carbohydrate loading" procedures during the hours and days before the event. It might be argued that for performances longer than three hours, it might be important to minimize unnecessary expenditure of the "quicker" fuel of stored glycogen, in order to minimize the need to substitute "slower" fuels such as stored fats. But it's probably more complicated than that.

Many proponents of this approach seem to assume that:

  • It's easy to know which moves do not (and could not even with better timing and preparation?) add net positive propulsive work versus which moves are obviously productive.

But for lots of moves, I don't see how this conclusion could be reached without some very careful analysis of both the primary move and its corresponding recovery move -- and many possible timing coordination with other moves and bodily configurations.

  • It will be straightforward for the athlete to find some other move to apply this energy, once the unproductive moves are eliminated.

But lots of athletes are already working hard on all the obvious productive moves they know about. Seems to me this method would do better to first identify positive alternatives, and only then focus attentional energy on eliminating something whose worst characteristic is "unproductive".

My observation:

  • Lots of winning athletes make moves which lots of observers claim are "obviously" unproductive.

beauty of form

My response:

  • "beauty of form" is a worthy goal. If you want to make into a competition, call it a "beauty of form" contest.  But don't call it a "race", and don't call it "SPEED-skating".

  • Different observers have different "eyes" for different styles of beauty.

  • There's nothing in the physics that seems to connect beauty with speed.

physics of speed and propulsion: equilibrium


Physics says that the forward speed of the person moving is the speed at which the total resistive force is equal to the total propulsive force delivered by all the muscle moves.

This is the "equilibrium" condition for speed. If the propulsive force is higher than the resistive force, the net extra force goes into accelerating the moving person to a higher speed -- until eventually a speed is reached at which the propulsive is not higher. If the resistive force is higher than the propulsive, the extra resistive force goes into decelerating the moving person to a lower speed. Normally there is only one particular speed at which the two kinds of force are exactly equal.

Resistive Force

Resistive forces are non-muscular forces which act on the self-propelled moving person, and tend to be include a component aimed opposite to the direction of forward motion. They include air resistance, gravity from hill steepness, and friction of sliding or rolling on the ground-surface.

A typical (simplified) formula for resistive force would be:

fR  =  (cf  +  sin γ) mP g  +  ca (vP + vH)2


fR  =  resistive force, overall total

cf  =  coefficient of friction for rolling or sliding on ground surface

sin  =  trigonometric "sine" function

γ  =  angle of any uphill slope begin climbed.

mP  =  mass of the moving Person

g  =  acceleration of gravity at the surface of the earth

ca  =  coefficient for air resistance.

vP  =  velocity of the moving Person

vH  =  velocity of any headwind opposing the Person's motion.

Actually resistive force can vary somewhat over the duration of each stroke-cycle, so actually that's a formula for something like "average" resistive force.

Most resistive forces are strongly dependent on forward speed -- and their resistance tends to increase positively with the speed of the person's forward motion. So a key reason we don't move much faster than we do is because it gets harder when we try to go faster. And it takes more physical Power, for two reasons: the Force is higher, the Velocity is higher, and Power equals Force times Velocity.

PP  =  fR vP

so for the case where there is no headwind, vH = 0, the typical formula for the physical Power which the person must exert to achieve a desire speed vP is:

PP  =  (cf + sin γ) mP g vP  +  ca vP3

Power determines key body-physiology requirements

Some important factors for muscle-powered propulsion are roughly proportional to this required physical Power:

  • oxygen which must be delivered from the environment to the muscle

  • chemical fuel which must be already present in or delivered to the muscle

  • heat which must be dissipated

For going faster, often we focus on propulsive force, but some of the biggest gains are from reducing resistive force: like the big difference in velocity on level ground between a recumbent bicycle and a standard racing bicycle -- because of the different in ca, the coefficient of air resistance.

Propulsive Force

I define propulsive forces as the muscular forces which (hopefully) have a component which is aimed in the direction opposite to the forward motion of the self-propelled moving person -- "opposite" because the force must push the ground backward, so that the ground will push the moving Person forward.

The force from a particular muscle move can be negative for propulsion -- if it transmits to the ground a component aimed toward the direction of forward motion. But since it's still a muscular force, I still include it in the "propulsive" category -- but note that it is negative for propulsion.

But note that a force can be exerted with a component aimed toward the direction of forward motion, yet have it transmitted to the ground aimed away from the direction of forward motion -- for example in the tricky physics of skating -- see the magic of skating.

It's important to analyze both the positive and negative aspects of both a pushing move -- and its corresponding recovery move -- over the entire stroke-cycle. Often one move of the pair is positive and the other negative, so it's important to know which direction it falls -- and if it's little positive or big positive.

The time-duration of most muscle moves effective for propulsion is much less than the whole stroke-cycle. So it's important to keep track of when each one starts and stops. Then the average total propulsive force over the whole stroke-cycle is a sort of "time and distance weighted" blend of all those positive (and negative) muscular forces.

The magnitude of the push-force exerted by the muscle is determined by what the muscle delivering can sustain repeatably based on the duration of the performance desired, the larger context of terrain constraints and event strategy, the frequency of moves per minute, the range-of-motion of the move, the muscular velocity of the move, the percentage of work versus rest for that move in the stroke-cycle, etc. -- see more on this under Biomechanics + physiology of muscular pushes.

what portion of exerted force is propulsive?

The proportion of the exerted force which is propulsive depends on how much of it gets transmitted to the ground (instead of being absorbed into body parts or equipment between the muscle and the ground surface) -- and the means of transmission, and at what angle the force is exerted and gets transmitted.

Let x be the unit vector in the direction of overall desired forward motion.

If the tranmission is multi-directional, such as through the foot in the motion of running, then the magnitude of currently propulsive force is

fP  =  fei x


fP  =  magnitude of currently propulsive force

fei  =  force vector exerted by the muscle move i

x  =  unit vector in the direction of overall desired forward motion.

If the transmission to the ground is in single specific direction, such as through the foot in the motion of ice skating, then the magnitude of currently propulsive force is:

fP  =  (fei n) (n x)


n  =  unit vector in the direction the horizontal push of the skate (which is perpendicular to the aiming-direction of the foot)

The push through the foot in skating also generates a "potential future" propulsive force whose magnitude is:

(fei n) (n y)


y  =  unit vector aimed horizontally sideways (away from the current push), perpendicular to the direction of forward motion x

How much of this "potential future" force can be used for propulsion depends on how long it is applied to the mass of the skater's body to build up how much sideways kinetic energy -- and how much of this kinetic energy can be converted and transmitted into future pushing force.

the magic of skating


x  =  (1, 0, 0);  y  =  (0, 1, 0);  z  =  (0, 0, 1)

n  =  (0.5, 0.866, 0) for skate aimed 30 away off from straight forward.

fei  =  ( 0, 100, 80) for push-force aimed straight out toward the side  (and partly downward)

then the magnitude of currently propulsive force is:

fP  =  (fei n) (n x=  (86.6) (0.5)  =  43.3

So a force exerted straight out toward the side is transmitted into a positive currently propulsive magnitude.

And some of this force exerted could also build up sideways kinetic energy which could be transmitted into propulsive force in the next skating-push through the other foot toward the other side.

Now suppose the skating push-force exerted is aimed partly forward from straight out toward the side -- so the exerted force actually has a component which opposes forward motion:

fei  =  ( 25.8, 96.6, 80) for push-force aimed 15 forward from straight out toward side (and partly downward)

then the magnitude of currently propulsive force is:

fP  =  (fei n) (n x=  (12.9 + 83.66) (0.5)  =  35.4

So the push-force exerted includes a push partly toward the forward direction, which seems to oppose forward motion of the person -- but it is transmitted into a positive currently propulsive magnitude.

Side effects

There are other factors which can be important for sustainable propulsion:

  • consumption of fuel (limited by total amount or rate of delivery)

  • dissipation of heat from muscular effort

  • removal of waste products from muscular effort.

  • repetitive motion stress on joints and ligaments

Sometimes moves must be made or muscular force must be applied to maintain a bodily configuration, though not propulsive:

  • support body weight

  • maintain balance

  • maintain sufficient friction against the ground surface to transmit propulsive force (very important for "classic striding" technique in cross-country skiing)

For purposes of this analysis, we will assume that any impacts from these factors have already been included in the specification of other constraint functions.

biomechanics + physiology of muscular pushes

Some points about muscular propulsion:

  • The total pushing force over the whole stroke-cycle comes from multiple muscle moves.

  • Many of the muscle moves are the product of multiple sub-moves or articulations.

  • Each muscle move is not made continuously, but only goes for part of the stroke-cycle, and has a starting and stopping time during the stroke-cycle.

  • Each move requires that the moving Person's body first be in a suitable configuration of relative angles among the joints, before it can be performed effectively. Usually the maximum range-of-motion or total work from the move is partly determined by the bodily configuration at the start of the move.

  • Recovery moves -- (almost?) every muscle move requires a "recovery" move to be performed before the same muscle move can be repeated.

This is because the moving Person's body configuration after the move is completed is different from what was required to start the move effectively -- so another rather different move must be performed to change the body configuration back to what it was at the start of the move.

Sometimes the "recovery" move also adds positive propulsive work, but in other cases its contribution is negative -- it cancels some (or all) of the propulsive work of the primary move. Analysis and design and timing of the recovery move is often critical for deciding whether a proposed muscle is suitable for inclusion in the "motion technique ensemble" for the moving Person to use for propulsion in a given situation.

  • The amount of force exerted by a muscle move usually varies through the time of its application, between the start and stop of the move.

  • The timing of one muscle move can overlap with the timing of another muscle move.

  • One muscle move could conflict with or reduce the effectiveness of another muscle move during their time of overlap.

  • There could be a "dead spot" time in the stroke-cycle where there is no muscle-move currently delivering effective propulsive force.


[ more to be added ]


physics of muscular pushes for speed


propulsive work

For each possible muscle move i, which is made during only a part of the stroke-cycle, from time t0i to time t1i, its incremental impact on propulsive work is (in simplified form):

WPi  =  ri τi(vθi, θ0i, θ1i , tc) Ti*P(t), v*θP(t), τ*P(t)] vθi(t) dt


WPi  incremental impact on propulsive work of muscle move i

  =  definite integral on time variable t from t0i to t1i

t  =  time since beginning of the current stroke-cycle

t0i  =  start time in the stroke-cycle for muscle move i

t1i  =  finish time in the stroke-cycle for muscle move i

ri  =  effective radius of muscle move i

τi ( ) =  torque function for muscle move i

vθi  =  "average" radial velocity for muscle move i

θ0i  =  start angular position for range-of-motion of muscle move i

θ1i  =  finish angular position for range-of-motion of muscle move i

tc  =  time of the overall stroke-cycle

Ti [ ]  =  propulsive Transmission effectiveness function for muscle move i

θ*P(t)  =  set of angle positions for all muscle moves of Person at time t

v*θP(t)  =  set of angular velocities for all muscle moves of Person at time t

τ*P(t)  =  set of torques for all muscle moves of Person at time t

vθi(t)  =  radial velocity of muscle move i at time t

Note that the moving Person does not usually get to choose the level of vθi(t) -- usually it is largely determined by many other variables in the moving Person's configuration and other muscle moves. 

resistive work

For each possible muscle move i, its incremental impact on resistive work is (in simplified form):

WRi  =  fRi[vθi, θ0i, θ1i, θ*P(t), v*θP(t), vP] vP dt


WRi  incremental impact on resistive work from muscle move i

  =  definite integral on time variable t from t0i to t1i

fRi [ ]  =  resistive force function for muscle move i

vP  =  average forward velocity of the moving Person.

equilibrium condition for speed

If we define FRavg as the expected average total resistive force against the moving Person, then the force equilibrium condition is:

FRavg vP tc  +  WRi  =  WPi


FRavg  expected average total resistive force against the Person

vP  =  average forward velocity of the moving Person.

tc  =  time of the overall stroke-cycle

  =  summation over all muscle moves, including recovery moves.

WRi  incremental impact on resistive work from muscle move i

WPi  incremental impact on propulsive work of muscle move i

Solving for speed gives:

vP  =  (WPiWRi) / (FRavg tc)


how to increase maximum speed

The formula above for equilibrium speed suggests several ways to increase speed:

  • Find additional specific moves which (together with their corresponding recovery moves) add net positive propulsive work.

  • Reduce the overall average resistive force (perhaps by finding a body configuration with lower air resistance.

  • Reduce the stroke-cycle time -- which is the same is increasing the turnover frequency or cadence -- perhaps by finding moves (or sets of moves) whose timing can be overlapped. Or . . .

Could try shortening the range-of-motion for a segment of a move (or set of moves) with Power is lower than average:

(WPiWRi) / (t1it0i)FRavg vP

provided that shortening this set of moves can also shorten the overall stroke-cycle.


[ more to be added ]


analyze propulsive characteristics of each muscle move


[ more to be added ]


central simple constraint: VO2max


trickiness of VO2max

not the true constraint for most performance durations

For very short periods it is possible to perform at a percentage higher than 100% -- so actually VO2max is not truly a fixed limit on muscular exertion.

But for performances of duration longer than 10 minutes it is surely not possible to sustain 100% of VO2max. So it's not a useful simple constraint for those performances either.

VO2max can change

VO2max can be changed with training over months and years. One problem for well-trained athletes to further increase their VO2max is this:  The capacity of the central cardio-vascular system responds so quickly and dramatically to the stress-stimulus of increased muscular demand for oxygen, that it becomes difficult for the athlete to set up demands on it which are greater than their previous training and event performances.

They cannot set up the oxygen-demand stress to increase VO2max, unless they first can develop and recruit more different muscles -- because they've already developed all the obvious muscles to their maximum oxygen-utilization capability.

Which is one of the objectives of this website: To find clever ways to use more different muscles.

Not clear what is "true" VO2max

It's not easy to determine what a person's "true" VO2max is. An experiment can be done with a particular mode of propulsion (e.g. pedaling on a stationary bicycle), but it cannot be certain that a person could not have demonstrated a higher VO2max number if the experiment had been done with a different propulsive motion (e.g. cross-country skiing with poling) -- provided the person did several months (or years) of training specific for that other propulsive motion.

Therefore, some practitioners say we should speak only of "VO2peak" -- the maximum measured for some particular propulsive motion. For an athlete who trains seriously for multiple different propulsive motions (e.g. swimming, bicycling, running), each motion could have a different VO2peak. But perhaps this triathlete's "true maximum" VO2max might be attained only in some fourth motion (e.g. cross-country skiing).


[ more to be added ]


practical implications



[ more to be added ]


more . . .


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