energy efficiency

A bit of a work out

While still reflecting on the way we use terminology so misleadingly within gait analysis it might be worth thinking a little about the concept of external work. It’s a concept that is even older than I am. Although previous workers (notably Fenn and Elftman) had used similar concepts it was Giovanni Cavagna who popularised it with his classic paper from 1963.  (Cavagna et al. 1963). The article starts with the sentence, “The work performed in walking can be considered as being made of two components, the internal work and the external work”. My response to this is that you can consider it like that if you want but you are likely to confuse people if you do!

Graphs from Cavagna's 1963 paper showing how horizontal components of speed and displacement are calculated from acceleration data. Note that his data was taken from an accelerometer worn on the body whereas it is more common these days for similar techniques to be used based on forde plate measurements.

Graphs from Cavagna’s 1963 paper showing how horizontal components of speed and displacement are calculated from acceleration data. Note that his data was taken from an accelerometer worn on the body whereas it is more common these days for similar techniques to be used based on force plate measurements.

Let’s be clear that there is no external work in walking. All the work required for walking is generated internally by the muscles. The result of muscles (and ligaments) exerting forces on the skeleton is that the foot exerts a force against the floor and generates the ground reaction (following Newton’s third law) but the ground reaction itself doesn’t do any work. It can’t. In order for a force to do work the point of application needs to move and the ground doesn’t move (well, not very often).

Whether its name is correct of not, the concept is important because it allows an estimate of the energy cost of walking on the basis of force plate measurements alone (cuts out all that nasty kinematics). The theory behind the calculations is generally presented as  straightforward but actually requires some quite subtle reasoning.

Although the ground reaction doesn’t do any work, it is a force applied externally to the body and will result in the centre of mass of the body being accelerated (Newton’s first law). If we measure the ground reaction we can thus calculate this acceleration and thus how the centre of mass is moving (its velocity and displacement).

Now if we wanted to move an equivalent mass through the same trajectory we could do so by applying an external force of the same magnitude and direction as the ground reaction directly to its centre of mass. If we did this then the point of application of this imaginary force would move and it would do work. Knowing the laws of physics it is reasonably easy to calculate what this work would be.

This can be taken as equivalent to the work that the muscles have to do to move the centre of mass, but it should be emphasized that the external force applied at the centre of mass is entirely imaginary, for the purposes of the calculation only. All the work is done internally by the muscles.

Of course this is one of those areas where people who understand the underlying concepts can cope with the fact that the name is wrong and get on with life … but I suspect that the terminology has the potential to be extremely misleading for those who don’t.

Additional note. It may also be worth being explicit that the muscles do other things as well as moving the centre of mass. They also move the segments with respect to the centre of mass and the work required to do this is not captured in the calculation outlined above. The calculation will thus always be an under estimate of the true mechanical cost of walking. It’s interesting that despite the extent to which these techniques have been used there have been very few studies of how much of an under-estimate, either for normal walking or for walking with pathology of different kinds.


Spoonful of sugar: a re-think

One of my earliest posts was about how efficient walking is and how little energy it takes to walk around. I illustrated this by the observation that it takes only the energy contained in two heaped teaspoons of sugar to allow an average adult to walk for a kilometre at a comfortable pace. After mentioning this as part of a tutorial I gave last month at the Gait and Clinical Movement Analysis in Portland several of us sat on chatting about whether this is a small amount of energy or not.

There is a problem in trying to think about how efficient walking is in that efficiency is generally defined as the energy output by a system divided by the energy that is input to a system.  The problem in relation to walking over level surfaces is that it doesn’t necessarily take any energy to move an object from one point to another if it is at the same height and moving at the same speed at the end of the movement as it was at the start. Think of a perfect wheel,  once we use a relatively small amount of energy to get it rolling, it will continue to roll along a level surface without requiring any energy input. If the energy output by the system is zero then it makes the calculation of efficiency a rather pointless exercise. Nothing divided by two teaspoons of sugar is nothing but so is nothing divided by one teaspoon of sugar or nothing divided by a hundred teaspoons. How can we get a handle on whether two teaspoons is a lot of energy or not?

One way might be to calculate the gradient of a slope we would have to be walking down in order for the loss in height to be provide the energy for walking rather than our bodies burning up food. Ralston’s classic paper of 1958 calculated the nutritional energy cost of walking (the amount of food energy that needs to be consumed) to be about 3.3 Joules/kg/m (assuming 1 cal = 4.186J) and more recent work that I’ve published agrees.  If that energy all came from a loss of potential energy  (mass x height x acceleration due to gravity) then it is quite easy to calculate that this would require a loss of 0.33m for every metre walked (=3.3/9.81). The gradient we would have to walk down would be 1 in 3 which sounds very steep to me.

Gradient 2

A slope with a gradient of 1 in 3

But things are not quite as simple as this. The efficiency with which food energy is converted to mechanical energy is estimated to be about 20% so the mechanical work that 3.3 J/kg/m represents is about 20% of the nutritional energy cost. This energy is thus only really equivalent to walking down an 1 in 15 gradient. It is also important to remember that about half of the gross energy cost of walking comes from the basal metabolism that is required to keep your body functioning regardless of whether you are walking or not. On this basis the effective gradient should perhaps be reduced even further to 1 in 30.


A slope with gradient 1 in 30.

So does this help? Well the 1 in 3 slope that we arrived at when just thinking about the nutritional cost is quite steep and perhaps serves as a reminder that the energy density of foods such as sugar is very high. We shouldn’t assume that just because we’ve got a a small amount of sugar that we have a small amount of energy. On balance, however, I think the 1 in 30 slope that arises when we take account of the basal metabolism and the efficiency of conversion of food to mechanical energy is a fairer reflection of how efficient walking is. This slope looks quite gentle and I think the overall conclusion that the walking is reasonably efficient is justified. The gradient isn’t however so small that it can just be ignored. The mechanical cost of walking appears to be the equivalent of raising the body mass by about 4cm more than is necessary for every stride (assuming a stride is about 1.3m long).

Plotting to convince

This post has been prompted by Mike’s comment on my last post. He pointed out two papers (Holt et al. 1991 and Minetti et al. 1995)  that have investigated the relationship between stride frequency and oxygen rate (per time) when walking at constant speed on a treadmill. The papers come up with the two graphs which I’ve included below (Holt et al. on the left, Minetti et al. on the right).

Minetti Holt

In many ways I’m more interested in how the data is plotted than by what the results actually are.

Both show a u-shaped realtionship but that on the left gives the impression of a rather broad curve with a poorly defined minimum whereas that on the right tends to suggest a much deeper curve with a well defined minimum. The graphs look very different to me to the extent that I might even interpret the data differently. I might interpret the left hand graph as suggesting that you can vary your stride frequency between about 0.9 and say 1.1 Hz without making much difference to oxygen rate. The message I might take out of the right hand graph is that oxygen rate is highly dependent on frequency with a clear minimum at a little under PSF+5.

But then I realise that the axes are quite different. From the Holt paper we find that PSF is 57 strides per minute (0.94 Hz) so the range from PSF -15 to PSF+15 is from 0.7 and 1.2 Hz and we also find that the average weight was  70 kg so the range of VO2 (0.8-1.4 l/min) corresponds to about 11.5 – 20 ml/min/kg. We can use this information ot overplot the graph of Holt et al. onto that of Minetti et al.:

Minetti holt

The data shows reasonable agreement given that the studies are not identical (and we have no way of knowing if the walking speeds were similar as this is not even recorded by Holt et al). What interests me is that conclusions are reversed. The data from Holt et al which appeared to show such a well defined minimum is even more broadly distributed than that from Minetti et al.

Conclusion one is that whilst there is a minimum in energy rate at about the preferred cadence this minimum is quite broad and with little change over quite a range of values either side of the minimum. Conclusion two is that choices in how you plot results can have quite a pronounced effect on how they are interpretted.


PS. For those of you who didn’t follow the comments yesterday this analysis is of oxygen rate with cadence at fixed speed is related to but different from the question of whether oxygen cost has a minimum value at self-selected walking speed which was the main focus of yesterday’s post.


Holt, K. G., Hamill, J., & Andres, R. O. (1991). Predicting the minimal energy costs of human walking. Med Sci Sports Exerc, 23(4), 491-498.

Minetti, A. E., Capelli, C., Zamparo, P., di Prampero, P. E., & Saibene, F. (1995). Effects of stride frequency on mechanical power and energy expenditure of walking. Med Sci Sports Exerc, 27(8), 1194-1202.

Walking in the groove

While I surfing the web doing a bit of background reading for last week’s post I came across this graph.

Ralston HJ (1958) Energy-speed relation and optimal speed during level walking. Int Z angew. Physiol. einschl. Arbeitsphysiol. 17 (8): 273-288.

Ralston HJ (1958) Energy-speed relation and optimal speed during level walking. Int Z angew. Physiol. einschl. Arbeitsphysiol. 17 (8): 273-288.

It’s another of the classic outputs of Verne Inman’s group, from Henry Ralston, and shows data for a healthy subject to support his hypothesis that we select our walking speed to minimise the energy cost of walking (the energy used to travel a certain distance). The hypothesis is so plausible that it has been almost universally accepted.

What interests me is that despite being so widely accepted I’ve never seen any suggestion of the mechanism through which we might achieve this. It’s a fairly basic principle of control theory that if we want to minimise any particular variable (such as distance walked for a given amount of energy) we need some way of measuring it. Thus it is very difficult to drive a car fuel efficiently if you just have a speedometer and a standard fuel gauge. If you add a readout to the dashboard telling you how much fuel you are using per kilometre travelled and the task becomes trivial. They should be compulsory in a fuel challenged world!

I’m not aware of any proprioceptive mechanism that would allow the brain to “know” how much energy it is using per unit distance walked. I can see that there are complex mechanisms regulating cardiac and pulmonary rate based primarily on carbon dioxide concentration in the blood which might allow us to sense how much energy we are using per unit time, but how can we possible sense how much energy we are using per unit distance. I’m not saying it’s impossible – the brain is a marvellous organ and it is possible that it integrates such a measure of energy rate (per unit time) with information about cadence and proprioception of joint angle and in order to derive a measure of energy cost (per unit distance). This is a complex mechanism however and certainly suggests that, as with so much in biology, whilst the basic hypothesis is extremely simple the mechanisms required to achieve this is far more complex than we might have imagined. As Ralston himself put it, “one of the most interesting problems in physiology is to elucidate the built in mechanism by which a person tends to adopt an optimum walking velocity such that energy expenditure per unit distance is a minimum”.

But this also makes me want to question the underlying hypothesis. Going back to the original paper (which you can read here), Ralston only produces data from one healthy subject and one amputee to support his hypothesis. I’m not aware of many others having explored the hypothesis on an individual level (the conclusion that the self-selected walking speed is close to speed of minimum energy cost for a sample does not mean that the relationship holds for individuals within that sample). I’d be interested to hear from readers of papers that have investigated this relationship in more detail.

The other point that Ralston made which is almost always overlooked is that the curve is “almost flat”. The curve only looks so steep because it has been plotted over such a wide range of values (from 0 through to 150m/s). Just looking at the data plotted I’d suggest that the speed can range from about  56 to 84 m/min whilst the energy cost remains within 5% of the minimum energy cost value. This is almost certainly within the range of measurement error for a variable such as energy cost. In other words the really remarkable thing about the energy curve is that it allows us to walk over quite a range of speeds without having a measureable effect on our energy cost. It is interesting that Ralston managed to make this point and suggest that we select walking speed to minimise energy cost in the same paper!

60 years of the determinants of gait: a misconception

The month of July 2013 marks the 60th anniversary of the publication of The Major Determinants of Normal and Pathological Gait by J B dec M Saunders, Verne Inman and Howard Eberhart.  This is a seminal paper in the history of gait analysis which was revered for many years and is the foundation of the description of normal walking in many text books.  More recently, however, it has come in for substantial criticism.

three determinants

The first named author, John Bertrand deCusance Morant Saunders, was a medically trained Professor of Anatomy at the University of California who was born in South Africa of Scottish descent. The story is that he needed his name on a paper to justify a trip to the Joint Meeting of the Orthopaedic Associations in London in 1952 and Inman and Eberhart obliged. There is little doubt that the ideas were those of Inman, a pioneering Orthopaedic Surgeon, and Eberhart,  an engineer. (Inman first met Eberhart when amputating his leg after a wartime accident at the time when he had been asked to establish the National Research Council Advisory Committee on Artificial Limbs. He invited Eberhart, originally a civil engineer, to join him and the partnership continued for the next thirty years).

Over the month I intend to write a series of posts celebrating this anniversary by looking at different aspects of the paper.  In this post I’d like to dispel one of the myths about the paper which is that it states that the aim of walking is to minimise the excursion of the centre of mass. In a significant review article, for example, Art Kuo (2007) writes “The six determinants of gait theory proposes that a set of kinematic features help to reduce the displacement of the centre of mass. It is based on the premise that the horizontal and vertical displacements are energetically costly”. 

An earlier paper by Ortega and Farley (2005) starts with an almost identical quote which drove the authors to train participants to walk with a nearly flat trajectory of the centre of mass. They then showed that it took nearly twice as much energy (oxygen) to walk a given distance with the flattened trajectory than with the normal trajectory. Gordon, Ferris and Kuo (2009 – who I think did the work earlier but published it considerably later than Ortega and Farley) conducted a very similar study and came up with essentially the same results. The introduction of that paper is interesting in describing how “at least a dozen text books have interpreted [Inman’s] work as meaning it is desirable to minimise or reduce COM movement during walking” and giving an overview of how the ideas have developed through these.

What is interesting though is that nowhere in the original paper (nor in the extended versions that have appeared in the three editions of the book Human Walking) can I find any statement by the  authors that minimisation of the COM movement is the aim of walking. What thy actually said was this:

Translation of a body in straight line with the least expenditure of energy may be achieved mechanically by the use of a wheel, but it is quite impossible by means of bipedal gait. The next most economical method would be the translation of the body through a sinusoidal pathway of low amplitude in which the deflections are gradual. Since force is equal to mass times acceleration and acceleration is a function of time, abrupt changes in the direction of the centre of motion compel a high expenditure of energy. In translating the centre of gravity through a smooth undulating pathway of low amplitude the human body conserves energy; and, as we shall see in considering pathological gait, the body will make every attempt to continue to conserve energy.

What they are proposing is that the body acts to ensure a smooth trajectory not necessarily one of minimal vertical displacement. They start off by describing compass gait, moving with fixed knee with no foot and the problem that they identify with this is that “at the point of intersection with the arcs, the abrupt change in the direction of the forward acceleration [I think they actually mean vertical component of velocity – RB] would require the application of a force of considerable magnitude”. This is actually extremely close to the hypothesis of the Dynamic Walking Group that one of the principal energy costs of walking is the requirement to redirect the centre of mass velocity during step to step transitions (Kuo et al. 2005) despite a contention that  their approach is the antithesis of Inman and Eberhart’s (see Kuo  2007). The six determinants proposed in the original paper are then strategies to smooth the trajectory of the COM but not necessarily to reduce it.

So where did the original and perfectly sensible views of Inman and Eberhart get distorted? Gordon et al. (2009) quote Perry (1992) as saying “minimising the amount that the centre of gravity is displaced from the line of progression is the major mechanism for reducing the muscular effort of walking, and consequently, saving energy”. Perry, of course, trained under Inman, and it may be that like so many pupils it is she that has misrepresented the ideas of her teacher. As an engineer myself, however, I’d take the personal side out. I’d see the original and valid ideas as indicative of the potential for progress when clinicians and engineers come together to address the challenges of clinical biomechanics. The misrepresented and invalid ideas appear when clinicians think they can go it alone!

That’s it for this post. I’ve emphasised one particular aspect in which I think the work has been unfairly criticised. In later posts I’ll look at some aspects where criticism may have been more justified as well as examining the popular appeal of the approach


Gordon, K. E., Ferris, D. P., & Kuo, A. D. (2009). Metabolic and mechanical energy costs of reducing vertical center of mass movement during gait. Arch Phys Med Rehabil, 90(1), 136-144.

Kuo, A. D., Donelan, J. M., & Ruina, A. (2005). Energetic consequences of walking like an inverted pendulum: step-to-step transitions. Exerc Sport Sci Rev, 33(2), 88-97.

Kuo, A. D. (2007). The six determinants of gait and the inverted pendulum analogy: A dynamic walking perspective. Hum Mov Sci, 26(4), 617-656.

Ortega, J. D., & Farley, C. T. (2005). Minimizing center of mass vertical movement increases metabolic cost in walking. J Appl Physiol, 99(6), 2099-2107.

Perry, J. (1992). Gait Analysis. Thorofare: SLACK.

Saunders, J. D. M., Inman, V. T., & Eberhart, H. D. (1953). The major determinants in normal and pathological gait. Journal of Bone and Joint Surgery, 35A(3), 543-728.