My posts on the equations of motion of the runner on Jan 16^{th} and Feb 6^{th} led to an intense discussion which included some very thought provoking comments by several readers, including Ewen, Simon, Robert, Mike, and Klas. In essence, the discussion led to the conclusion that the calculations themselves provide an accurate account of the mechanical energy costs of the braking that is inevitable when the point of support is in front of the centre of gravity (COG), and the cost of elevating the body to become airborne. However, these costs are not the only costs that need to be considered. The other main mechanical cost is the cost of repositioning the limbs relative to the COG, In addition, possible variations in the efficiency of conversion of metabolic energy to mechanical energy, and the efficiency of recovery of energy via elastic recoil must be considered. Furthermore, factors such as wind resistance and variation in the profile of the time course of the pressure that the foot exerts upon the ground (and the opposing ground reaction force, GRF) should be borne in mind.

A complete account of the energy costs of running needs to take account of all of these factors. I believe it is possible to deal adequately with effects of wind residence and variation in the profile of the time course of GRF, and I plan to do this in future posts. I am confident that these factors play only a relatively minor role under many circumstances. Unfortunately, variation in the costs of repositioning the limbs; the efficiency of metabolic to mechanical conversion; and the efficiency of elastic recoil are difficult to estimate precisely but would be expected to play a key role under some circumstances. Nonetheless, I believe that for the range of time on stance, cadence and speed that I considered in my calculations, the changes in braking costs and the costs of elevating the body that are achieved by adjusting cadence and time on stance the most important factors to consider. A full justification of this claim would require more detailed information about repositioning costs, efficiency of metabolic to mechanical conversion and elastic recoil than are currently available. In future posts I will also review what is known about each of these factors in detail. My goal in this post is to provide an outline of why I think that variation in cost of braking and elevation of the body are the most important issues in the circumstances discussed in my posts on 16^{th} Jan and 6^{th} Feb, and furthermore, to provide an indication of the range of running speeds over which my conclusions likely to be valid.

**Repositioning costs**

The largest of the repositioning costs is the energy required to move the leg forward relative to the COG during the swing phase. Muscles must do work accelerating the foot from a stationary position on the ground to a speed approaching twice the running speed by mid-swing, to allow the foot to overtake the torso and get ahead of the COG by foot-fall. Factors such as elastic recoil of the hip flexors which were stretched in late stance will contribute to the forward propulsion of the leg. After mid-swing, the leg decelerates so some of the energy imparted initially might be recovered as the leg pulls the torso forwards. Nonetheless, due to inefficiency neither elastic recoil of the hip flexors nor the momentum of the swinging leg will meet all of the cost. We can obtain a crude estimate of the magnitude of repositioning costs by applying Newton’s laws of motion estimate the mechanical cost of accelerating the leg forwards during the first half of swing.

The work that is done in accelerating an object is proportional to the integral of force times velocity over the time period for which the force acts. If we assume that the acceleration is uniform, it can readily be demonstrated using Newton’s laws, that the work done is proportional to the square of the running speed and inversely proportional to the duration of the swing. Thus the repositioning cost will increase rapidly as running speed increases. It will also increase as cadence increases since swing time decreases as cadence increases, assuming a constant proportion of time is spent on stance (which is the case if the peak vertical GRF is fixed). Furthermore, at fixed cadence, swing time decreases as time on stance increases, so increasing stance time will result in greater repositioning cost.

For the situation considered in my post on Feb 6^{th}, in which cadence increased from 180 to 200 steps per minute, while both velocity peak vGRF and hence proportion of timer spent on stance remained constant, the repositioning cost would be expected to increase by 11 per cent (20/180). For the situation considered in my post on Jan 16^{th}, running speed averaged over the gait cycle remained constant and cadence remained constant at 180 steps per minute, while peak vGRF/Kg increased from 2g to 4g. Time on stance decreased from 262 milliseconds to 131 milliseconds while swing time increased from 404 milliseconds to 535 milliseconds. Thus, repositioning costs would be expected to decrease by 32% (131/404). Although the assumption of uniform acceleration is an approximation that would make any estimate of actual energy cost unreliable, the estimate of the proportional change is likely to be a reasonably reliable guide for our present purpose. The most important issue is the direction of change in repositioning costs: namely at constant cadence, the repositioning costs decrease as peak vGRF increases and stance time decreases; while at constant vGRF, the repositioning costs increase as cadence increases.

What proportion of the total mechanical costs can be attributed to repositioning the limbs when running speed is 4 m/sec? As we have seen repositioning cost increase as the square of the running speed. This was confirmed by Cavagna and Kaneko (J. Phy8iol. (1977), 268, pp. 467-481) by direct measurement of the motion of the limbs recorded on cine films. Furthermore, C&K demonstrated that for runners who were running using their preferred running style at various speeds, that the repositioning costs were equal to the sum of braking and elevation costs at a speed of 20 Km/hour (5.5 m/sec). At 4 m/sec, the cost of repositioning the limbs was 37% of the total mechanical costs. Thus, when vGRF is kept constant while cadence increases from 180 to 200 steps per minute, the repositioning costs would be expected to increase the total mechanical work costs by about 4% (11% of 37%). In my computation presented on Feb 6^{th}, I demonstrated that the combined cost of braking and elevation diminished by about 10% (ie about 6.3% of total mechanical costs) as cadence increased from 180 to 200 steps per minute (at speed of 4 m/sec and peak vGRF/Kg = 3g). Thus the gain in mechanical efficiency achieved by increasing cadence is only a little greater than the added repositioning cost. It is clear that at speeds much faster than 4 m/sec, the gain in mechanical efficiency obtained by increasing cadence is likely to be obliterated by the increased repositioning costs. On the other hand, at slower speeds the gains from increasing cadence would be expected to be appreciable. Furthermore, at lower vales of peak vGRF which are associated with longer times on stance, the gains from increasing cadence would also be greater. So, in summary, at a speed of 4 m/sec and vGRF/Km = 3g, a small gain in mechanical efficiency would be expected when cadence increases from 180 to 200 steps per minute. However at higher speed or higher peak vGRF the gain from reduced braking would be offset by the increased repositioning costs. (Although I have not done the relevant calculations, even at 5.5 m/sec, where reposition cost is equal to the sum of braking and elevation cost, a net gain might be expected from increased cadence but it would be very small). In contrast, at lower speeds and/or lower peak vGRF, worthwhile gains in efficiency might be expected as cadence increases.

In the situation considered in my post on Jan 16^{th}, there was a 21% decrease in the braking and elevation costs as peak vGRF/Kg increases from 2g to 4g at constant cadence of 180. Based on Cavagna and Kaneko’s data, this represents a 13% saving in total mechanical cost. As we have seen in the above estimate, repositioning costs will be expected to decrease by about 32%. Since the C&K data indicate that repositioning cost are 37% of total mechanical costs at this speed, the reduced repositioning cost would be expected to produce a 12% saving in total mechanical costs. Thus when vGRF increases (and stance time decreases), the gain in efficiency for reduced braking is augmented by a gain of similar magnitude from reduced repositioning costs. At higher speeds, an even greater proportional gain in efficiency would be expected from increasing vGRF.

Although the numbers employed in these calculations are only approximate estimates, the general conclusions are likely to be valid. At 4 m/sec, increasing vGRF (and decreasing time on stance) at constant cadence produces an appreciable gain in efficiency due to reduced braking costs accompanied by reduced repositioning costs. Increasing cadence from 180 to 200 steps /minute produces only a small improvement in efficiency due to the counter-productive increase in repositioning costs. At higher speeds, the gains from increasing vGRF would be expected to be even greater, while the gains from increasing cadence would be minimal. In contrast at lower speeds the gains from increasing cadence would be expected to be appreciable.

**Effect of increasing vGRF at very slow speeds**.

As discussed in my post on Jan 16, as vGRF increases at constant cadence, braking costs decrease while elevation costs increase. At 4 m/sec, the gain from reduced braking cost is substantially greater than the extra elevation cost, so the combined mechanical cost of braking and elevation decreases as vGRF increases. However, at very slow speeds, the distance travelled while on stance is very small and the work that must be done to compensate or braking is much less, so braking costs are a lesser proportion of total mechanical costs. However, elevation costs (per step) for a given peak vGRF are almost independent of speed. (This emerged from the equations of motion and was confirmed experimentally by C&K.) This at very low speed, the combined cost of braking and elevation will actually increase as vGRF decreases. For example, in my response on 27^{th} Feb to a comment by Klas on my post of Jan 16^{th}, I presented results demonstrating that at a speed of 2.5 m/sec, combined cost of braking and elevation is actually greater when vGRF/Km =4g compared with 2g. Furthermore, at such a low speed, repositioning cost are very small. Therefore, at speed as slow as 2.5 m/sec, increasing vGRF produces no appreciable gain in mechanical efficiency.

I should also be noted that even at higher speeds, once stance time becomes extremely short, braking costs will be low compared with elevation costs and further reduction in time on stance will result in an increase in the combined cost of braking and elevation.

**Efficiency of conversion from metabolic to mechanical energy**.

The efficiency with which muscle contraction converts metabolic energy to mechanical energy is typically around 20% or even less in some circumstances. The largest contribution to this is the inefficiency of the biochemical process by which fuel is burned to produce the energy molecule ATP which proves energy to the contractile machinery of the muscle fibre. This process has an efficiency of only 40%. Unfortunately, no adjustment of running style can improve this biochemical inefficiency. However the efficiency of processes by which the molecular machinery within muscle fibre generates force is potentially amenable to change. Muscles contract by a ratcheting action between filamentous actin and myosin molecules within the muscle fibre. The efficiency of this ratcheting depends on the rate of shortening of the muscle. There is a certain fairly narrow range of contraction speeds at which the interaction between actin and myosin is optimally efficient.

Efficiency falls away rapidly when contraction speed is less the optimal range, and falls away somewhat less rapidly as contraction speed increases above the optimal. Different fibre types have different optimal speeds. As might be expected, slow twitch (type 1) fibres are optimally efficient at slower speed of contraction than fast twitch (type 2) fibres. The optimal contraction speeds for these two fibre types differ by a factor of about two. There appears to be a neural mechanical hat ensures that type of fibres that are recruited for a task depends on the demands of the task. Hence, at least for a professional athlete who has the opportunity to train whichever type of fibre is most relevant to his/her event, it would appear that the best strategy is to train the fibres that are most suited to achieving optimal mechanical efficiency. Maybe a recreational runner might be better advised to adjust factors such as peak vGRF to match the fibres that are available for the task.

Furthermore, the efficiency of metabolic to mechanical efficiency conversion diminishes as a muscle becomes fatigued (C.J.Barclay, Journal of Physiology (1996), 497.3, pp.781-794). Therefore, at least for a recreational runner, it might be better to adjust vGRF to a somewhat lower value than that which provides optimal mechanical efficiency, so as to increase the recruitment of the more fatigue resistant slow twitch fibres. The tendency for marathon runners to increase time on stance in the later stage of the race might reflect the need to rely almost entirely on slow twitch fibres in the later stages of the race.

In summary, it seems to me that preferred strategy is to train to produce adequate fatigue resistance in the fibres that are best suited to achieving optimal mechanical efficiency. However if one has less opportunity to train, or when racing over a distance that is longer than usual, it might make sense to increase time on stance to maximise the efficacy of conversion of metabolic energy to mechanical work despite some loss of mechanical efficiency.

**Elastic recoil**

The elasticity of tendons increases as the rate of application of force increases, so in general, the efficiency of elastic recoil of the tendon itself would be expected to be greater at shorter times on stance, though the as the rate of application of force increases a plateau would eventually be reached. However, perhaps more important than the plateau at high loading rates is the fact that recoil is a product of the concerted action of muscle and tendon. Tension is only created if the muscle contracts as the muscle-tendon unit is stretched. Therefore, if the rate of application of force is potentially too great for the muscle to bear without damage, it is likely that a protective mechanism will limit the amount of tension that is developed. Thus, it would be expected that the efficiency of elastic recoil will increase as time on stance decreases, but beyond a certain point the strength of the muscle contraction will cease to increase, and tension will no longer rise in proportion to the applied force. Thus elastic recoil will capture a smaller proportion of the energy of impact. I do not know of any measurements that establish the rate of loading that achieves maximum efficiency of elastic recoil during running. However, as in the above consideration of metabolic to mechanical conversion efficiency, it would appear that the ideal strategy for optimal efficiency is to develop muscle strength to the level required to cope with the loading rate required to give maximum mechanical efficiency.

These considerations suggest that the optimum strategy is to develop both strength and fatigue resistance of muscle fibres to a sufficient degree to allow the achievement of optimum mechanical efficiency. However in practice this might not be feasible, especially for recreational runners. In such situations it might be more efficient to adopt a somewhat longer time on stance even it this results in sacrifice some mechanical efficiency.

**Conclusion**

At running speeds around 4 m/sec or higher, the greatest mechanical efficiency is likely to be achieved by aiming for a relatively short time on stance, achieved by employing a greater peak vGRF. Furthermore, increasing cadence from 180 to 200 steps per minute would also be expected to produce a small some gain in efficiency, but the increased cost of repositioning the limbs nullifies some of the potential gain from reduced braking cost. At higher speeds, this antagonism of the potential benefit of increased cadence becomes even more marked.

The increased in peak vGRF required to achieve a shorter time on stance (at constant cadence) comes the price of greater stress on the muscles. At least for the recreational runner, and perhaps even for professional athletes running very long distances, it might be preferable to sacrifice some of the potential gain in mechanical efficiency by employing a somewhat longer time on stance.