Monday, February 13, 2017

Actionable Intelligence for Running Part 4 : Pacing by Power & Running Effectiveness

In Part 3 of this series, I continued looking into Leg Spring Stiffness (LSS) relationship with speed and time. I found that LSS is a function of speed and while it has been considered by some in the running community to be linearly increasing with speed, I found no evidence of strict linearity. 

Some limited data from long runs suggested that  LSS may also drop with duration, hinting that running form was being affected. That information is subject for further confirmation from new runs.

This post focuses mainly on running power. LSS, ground contact time, VO, ground reaction force, swing angle etc are all biomechanical ingrediants of what makes power. Power is an input to create propulsive speed. 

Many weeks ago, one of the questions I had on my mind was whether pacing by a powermeter is a valid approach. In cycling time trials, it is common for athletes to pace by a known power as a percentage of critical power (CP). Could running adopt that strategy?

To test this, I ran what is billed as the "World's Fastest Half Marathon", the RAK Half in Ras al Khaimah, U.A.E on February 2017. Being a flat out race with little undulating terrain, it was a perfect ground to inspect running power. I completed the half in a net time of 1:50:49, the reasons for which are below. 

Prior to this run, I had no idea of the correct value of my CP since I hadn't bothered to do any of the track based tests that Stryd advised. However, I had an idea of estimated CP from a multicomponent exponential model fitted to all the past runs with power since January 1, 2017. 

CP from the model in the immediate week before the RAK half was approximately 207 Watts.

To test whether pacing by power was a valid approach, I had to run at what I call "suicide pace", which is an unsustainable power output greater than the estimated CP. This had to be mostly by feel or RPE (rate of perceived exertion). 

The test protocol is below and deceptively simple. 

Test Protocol

1. I chose my "suicide pace" as around 230 Watts average.

230 ÷ 207 = 1.11 Intensity Factor.

2. I switched the watch to power mode and ran by knowledge of power alone. No speed, no heart rate, no duration,

3. I ran till certain 'death', i.e complete fatigue. I then limped to finish line, plugged the computer to post process the data.

The intelligence I gained from this exercise is, as in previous posts, summarized in blue italics.

Pacing By Power Insights

The following plot shows power and speed as a function of duration.

Fig.1 : Time dependant running power (black) and speed (yellow) for the RAK half.

The following are the insights I gain from the data :

1. Power tracked running speed remarkably well. Ups and downs in power followed ups and downs in speed, except for data regions showing spikes in speed.

2. A regression analysis on speed vs power from my RAK run suggests that an increase of 1 Watt translates to an increase in running speed of 0.0134 m/s = 0.03 mph. In other words, a 1 mph increase in speed demands a 33 Watt increase in power. 

3. About 57 minutes into the race, another tracker of effort - heart rate - crossed into my anaerobic zone, inviting fatigue due to lactate metabolites accumulating in the body. Plot overlayed with HR data is below, showing the exact point at which HR crossed into my red zone.

Fig.2 : Power and speed overlayed with heart rate for the duration of the race.

4. For the next 27 minutes, I was in pure purgatory. Pace kept decreasing steadily. At precisely 1:23:00 which I call Point of Exhaustion, I had to drastically decrease my pace to gear down HR. Essentially my test was over and from hereon it just an effort to get across the finish line. 

Stats for the first 57 minutes were as follows :

Duration = 57:00
Distance = 7.6 miles, 12.23 km
Pace = 07:28  min/mile, 4:39 min/km
Work = 804 kJ
Average Power = 235 Watts
Average Heart Rate = 195 bpm (Spot on at Threshold)
Average IF = 1.121
GOVSS = 66
Gradient = 0.1%
Average Step Rate = 185 SPM.

5. After feeding the data into the post processor, the time distribution of watts pins down the power target I should have honed into for the Half.  This is shown in Fig. 3. Had I geared down wattage to a range between 205-210 Watts, I'm led to believe that with current fitness level, I could have run the distance without issues. I also think this establishes CP far better, because when you feed good race data into CP models, you get a reliable output. More data there is, better reliability in the figures. I remain suspicious of power and pacing zones established outside of racing.

Fig. 3 : Time distribution of power readings. Zones established from Jim Vance methodology.

6. I believe running power helps quantify the demands of training and racing. It makes it possible to analyze and score training data based on Intensity Factor and Normalized Power, a 30 second rolling average algorithm introduced by Andrew Coggan and supported by Annan (2006) . 

7. Since power is Joules over second, it also accurately pinpoints the energy expenditure of a running race, giving clues as to why a runner 'bonked' for example. Power can be overlayed onto a course map by GIS techniques and used to pinpoint energy expenditute and power intensity on a specific stretch of the course. An example shown below in Fig.4, giving me some great data for one of the defining stretches of the RAK Half Marathon. 

Fig.4 : Segment of RAK Half course highlighted in red shows the power and energy requirements of running the 9 mile stretch of road, which happens to be a deciding point in the race.  

8. Running with power may also allow the study of how biomechanical adjustments translate to improvements in running. For example, it is possible to inspect if one style of running leads to better running effectiveness than another. I would want to look into these in future posts. Running effectivess is discussed below.

Running Effectiveness (RE) Insights

With running power data, one can analyze the ratio of speed (m/s) to power/weight ratio (watts/kg), effectively defining an achievable amount of speed for a given unit of power to weight ratio. Folks are also finding from track runs that finishing times among young runners correlate very strongly with RE. 

I calculated RE for the RAK half and the following are insights I gain from the RE data. (Please note I assumed that my weight was a constant 65kg for the duration of the race).

1. My average running effectiveness for the entire duration of the RAK Half was 1.00194  m/s/W/kg. The plot of RE and speed vs duration is shown below, displaying that even though speed dropped off, the trend of RE was more or less constant. 

Fig.5 : Power and speed data overlaid with calculated RE for the duration of the race.

2. The Spearman ranked correlation co-efficient between speed and RE is +0.334. A linear regression between just the raw values of speed vs RE from the run suggests that an increase of 1 kph in speed demands an increase in RE of 1.25 m/s/W/kg. 

3. One interesting technique of analysis is to linearize power, speed and running effectiveness. This is done by expressing each as a ratio of their respective averages for the duration of the race and multplying by 100 for a percentage. The linear trend lines of these 3 curves are shown in the plot below. 

Fig. 6 : Linear trend lines for relative speed, relative power and relative RE vs time. 

The plot shows linear trends for relative power and relative speed dropping off from start to end of race, perhaps suggesting the effect of fatigue.  However, linear trend for relative RE stays constant. From this, I gather that for the duration of a half marathon, the amount of speed you can produce for a set amount of watt/kg stays more or less constant. It is possible, however, that a longer duration race, such as a marathon or an ultramarathon, shows a more clearer drop in RE with time. In other words, fatigue may be more pronouned during long races that it also shows up in the RE data. This is something I will continue to look at in future.

Your comments welcome!

NOTE: My half marathon PB when I was younger is 1:40:51, done on hilly terrain. Link for results.

Saturday, February 11, 2017

Actionable Intelligence for Running Part 3 : Leg Spring Stiffness & Speed-Time Relationships

LSS is a theoretical concept under the linear spring mass model of running, which has proven to remarkably explain several characteristics of running. Mathematical models teach first principles. If most of what you want to know can be explained by a simple model, the additional time spent on increasing the complexity of modeling is not justified.

In Part 2 of this series, I found that change in Ground Contact Time (GCT) is a better explainer of change in Leg Spring Stiffness (LSS) than is change in cadence (step frequency or SF). 

In this post, I continue looking into LSS variation with running speed for both treadmill runs and outdoor runs on predominatly asphalt. Running subject is me again. The "intelligence" gained from this post is summarized in blue italics. Please visit Part 2 for abbreviations used in the writeup.

LSS and Speed Relationship for Treadmill Runs

Around 12,527 datapoints were collected over strictly controlled treadmill runs executed on a Cybex 770T from January 1 - February 9, 2017. The data was cleaned by eliminating null values of LSS; my Stryd footpod is expected to give such data from time to time, mostly during the start of runs. The LSS vs running speed plot is shown below.

Fig.1 : Scatter plot of Stryd measured LSS vs running speed for treadmill running. Sample frequency = 0.5 Hz.

The question is whether the data suggests the relationship is linear or not. The Excel based R^2 value of 0.7902 did not inform me correctly. A better method was to use a correlation test. 

A Ryan Joiner test to test the null hypothesis that the data was a random sample from a normal distribution. The probability plot of LSS from this dataset is shown below. Since the probability line is not straight, the RJ value < 1 and p-value was  .001

Fig.2 : Supplementary plot showing non-normality in LSS data collected during treadmill runs.

Next, a Spearman's rank order correlation test for non-normal data was done. The correlation coefficient was 0.104, p value = 0. This told me that the data has a monotonically increasing relation and that the probability of seeing the observed relationship was 0 if the relationship was anything other. The fact that the sample is large may show that Spearman's rank order correlation of 0.104 has high power. 

Fig. 1 also shows presence of outliers - datapoints of high LSS for the same magnitude of runs as the majority of parent data. What's going on here? When I peer into just these points, I find evidence that high LSS values (13kN/m or greater) can belong in either a high cadence region or a low cadence region. 

The low cadence region 1 consists of high vertical oscillation (VO) and high GCT. The high cadence region 2 consists of low VO and low GCT. After inspecting these subsets of data individually, region 1 was found to correspond with low speed strides I did on the treadmill (for want of some variation in a long run). Region 2 corresponded with high speed, high cadence running. These two regions are shown below in Fig.3.

Fig.3 : Plot showing two regimes where LSS data was found to be greater than 13 kN/m. Region 1 is low SF region with high VO and GCT, while region 2 is high SF region with low VO and GCT.

The fact that leg stiffness is high at high stride frequencies is consistent with literature. I didn't find confirmatory evidence in research for high values of leg stiffness at low step frequencies and high GCT. However, it logically makes sense that one would have to neuromechanically adjust leg stiffness to cushion a higher degree of fall from greater VO. I'll keep looking in future for data within region 1.

LSS and Speed Relationship for Outdoor Runs

I also looked for the relationship of LSS with speed for outdoor runs executed on predominantly paved surfaces. 10,164 datapoints were inspected from the period January 1 - February 9, 2017 and found to show a near monotonic relationship between speed and LSS. The Ryan Joiner test on this dataset rejected the null hypothesis that the data was a random drawing from a normal distribution (.001

Fig.4 : Scatter plot of Stryd measured LSS vs running speed for outdoor runs. Sample frequency = 0.5 Hz. This dataset includes a run done at -4 deg C ambient temperature when the running surface author encountered was covered with thin ice.

A negative correlation co-efficient was somewhat puzzling to me. I have to point out that 2,347 of these datapoints in the sample corresponded to a run done in the UK when ambient temperature was minus 4 degree C. There was still icy frost on the ground. During this run, I was trying consciously to stay upright and not slip on an unlit running path besides the Thames river at 6 in the morning. Neuromechanically, this might mean I could have changed leg stiffness to maintain center of gravity. That leg stiffnesses are adjusted by human runners based on type of running surface was shown by Farley  

Following that theory, I eliminated these 2,347 datapoints from the dataset and followed through with the rank ordered Spearman test on the rest of the 7,736 datapoints. The correlation coefficient was 0.022, p-value = 0.057. This told me LSS was typically monotonically increasing with running speed and within this dataset, the probability that it is a chance event is small. 

The takeaway : Given a more or less fixed step rate, leg spring stiffness is correlated with speed. The data shows that leg spring stiffness monotonically increases with speed. Majority of LSS values can be in a tight window, for example for all my runs they were typically between 10-12 kN/m. 

A cautionary note is that higher than typical LSS values may be exhibited during periods of slow speed, low step rate running where vertical oscillation is high (such as during strides). It may also be exhibited during periods of high speed, high step rate running where vertical oscillation is small. It is important to survey the behavior of these connected parameters when trying to assess why LSS numbers are higher than your typical values. When such data are accounted for in outdoor runs and indoor runs (running motion oddities due to the effect of "slippery" icy surfaces for example), the relationship between LSS and running speed is appreciably monotonic positive. 

LSS vs Speed and Time In Endurance Runs Greater than 1 Hour

I've not done too many distance runs greater than 1 hour 20 minutes. That said, there are two runs of between 1:20:00 - 1:40:00 which I thought could be inspected to see if long duration running and it's accompanying fatiguing aspects could translate to decreases in leg spring stiffness. The two runs with speed and LSS are displayed in Figs. 5 & 6. Plot captions indicate the conditions under which they were done. 

Fig.5 : 1 hz data from endurance run #1 executed in Abu Dhabi on February 3, 2017 under 17 deg C ambient temperature and 21 kph winds (around Beufort Scale 6). Number of sample points = 2,962. Plot shows LSS, GCT and speed variation with time. Mean LSS = 10.233 kN/m, mean speed = 3 m/s, mean GCT = 247.22 ms. 

Fig.6 : 1 hz data from endurance run #2 executed in Staines-on-Thames on January 23, 2017 under -4 deg C ambient temperature and low winds (around Beufort Scale 6). Number of sample points = 2,448. Plot shows LSS, GCT and speed variation with time. Mean LSS = 10.54 kN/m, mean speed = 2.67 m/s, mean GCT = 262.83 ms.

In Fig.5, the Spearman rank ordered correlation between duration and LSS showed a correlation coefficient of -0.337, p value = 0. In Fig.6, the Spearman rank ordered correlation between duration and LSS showed a correlation coefficient of only -0.094, p value = 0. The relationships in both these runs between duration and LSS was mildly monotonic decreasing.

What is interesting is that the faster, warmer, longer duration run in windier conditions shows a stronger negative correlation between LSS and duration than does the somewhat shorter, colder and slower run in cold temperature.

A contour plot of LSS, speed and duration for the run from Fig.5 is shown below. It indicates that there is a cluster of low LSS values in the range 5.0-7.5 kN/m corresponding to high speed and high duration. This is interesting and I need to concentrate on these instances both within the data and in reality to understand what is going on. For example, it hasn't been uncommon for my left feet to exhibit fatigue from long duration running.

Fig.7 : Contour plot of LSS vs Speed and Time in a long run on a windy day in Abu Dhabi on January 23, 2017

However, due to the lack of further long runs and lack of high intensity data, I cannot fully elaborate on fatigue and it's effects on LSS. It will be something I will keep a watch on. Perhaps introducing power and form power into these metrics will form a clearer picture.

The takeaway : A preliminary analysis of two long duration runs suggest there is a monotonic decreasing relationship between duration of run and LSS. It perhaps is likely that changes in running form and/or fatigue can influence LSS as evidenced by the contour plot. This aspect needs a much more comprehensive treatment with other markers of performance. 

In the next post, I will move to power based metrics and see how indoor and outdoor runs compare. I will also inspect my power data from the recent RAK half marathon and see how power based pacing helps in duration specific stamina and how that methodology compares to speed based pacing. Stay tuned.

Saturday, January 21, 2017

Actionable Intelligence for Running Part 2 : Effect of Step frequency on Leg Spring Characteristics for Treadmill Running

In Part 1, I reviewed research literature on the relationship of inverse ground contact time, 1/tc, to the metabolic cost of running. Further, because the leg stores and reuses elastic energy, spring mass models have been used quite extensively since the 1960's to describe it's spring characteristics. 

In this installment, I will investigate the relationships between leg spring characteristics against variable step frequency and variable ground contact time in two tests. I will use two wearable electronics - a Polar V800 and a Stryd gen 2 footpod - along with a treadmill and an Android phone. Force plates, kinematic arms or 3D motion capture systems were not used.   

A point to note is that the human leg is not a pure linear spring due to various muscoskeletal complexities, therefore the estimated stiffness variables will be preceded with the word "effective" in this writeup while device native metrics are called as is. 

Lastly, as the test was done on myself, there could be an effect of human error and testing inexperience shown in this study. When possible, I've tried to introduce randomization in the test protocol to equalize the effect of external factors not accounted for in the experimental design. You are free to mail me pointing out any flaws in understanding.

Commentary to the findings (or the "intelligence") is highlighted in blue. 

Abbreviations Used

Calculations Used

Read Part 1. Or ask.

Subject and Equipment Used

Test Protocol

1. 5 min warmup at 2.8 m/s pace.
2. 2 min run at 3.3 m/s pace to determine fcSF. 
3. Calculate 6 test SF values from 30% reduction to 30% increase in 10% increments : 0.7fcSF, 0.8fcSF, 0.9fcSF, +1.1fcSF, +1.2fcSF, +1.3fcSF.
They also also called SF-30% ,SF-20% , SF-10% ,SF10% ,SF20% & SF30%
4. Part A (Variable GCT - Variable SF test) : Randomize the order of the above and complete 6 trials of 2 min each at 3.33 m/s. 2 min recovery in between. 
5. Practice a running motion for self-imposing short GCT and long GCT. 
6. Part B (Variable GCT - Fixed SF test) : Complete 2 x 3 min randomized trials strictly at fcSF and at a pace of 3.33 m/s, one with short GCT and one with long GCT.
7. Consolidate data for post processing within a statistical computer package.

Note : 3.33 m/s corresponds to 12 kph on the treadmill and 8 min/mile. A cadence of 180 SPM corresponds to 3 Hz. 

I. Results of Test Part A 

The fcSF during the initial 2 min trial run was 177 SPM which is 2.95 Hz. 
Corresponding GCT was 235.5 ms. 
LSS at fcSF = 10.2 kN/m.

Calculated values of SF for the 6 trial runs were as follows :

626 data points were consolidated from this test. 

Data is plotted below.

Fig.1 ; Spread in data for Part A for each of the 6 SFs.

FIG 1 : Box plots of FT and GCT at the 6 different SF's are shown relative to fcSF (marked with a dotted reference line). 

The data corresponding to 2.065 Hz or
SF-30% has a large spread because I had difficulty trying to main such a low cadence, having to hold the sides of the treadmill for support at times. This plot has been made after cleaning up a lot of null values within the SF-30% dataset. Inspite of this, there still is great spread relative to data at other SFs.

The general takeaway is that ground contact times fall steadily with increase in step rate and rises with decrease in step rate. One would expect aerial times to be high in the low cadence region, but the data seems to show an inflection point at fcSF of 2.95Hz. In the increasing cadence region, FT tends to decrease.  

Fig.2 : Scatter plot showing % change in LSS vs % change in GCT during 
the variable GCT - variable SF test.

Fig.3 : Scatter plot showing % change in LSS vs % change in SF during 
the variable GCT - variable SF test.

FIGS 2 & 3 :  Decreasing changes in GCT obtained from the variable SF test correspond to positive changes in LSS. A simple linear regression shows R^2 value is 0.79. The Pearson correlation test of both yields -0.889 (P < 0.05).  

Problematic, however, is the fit of SF variation with LSS. Some of the noise in the left hand side is most likely due to the improperly executed low cadence test at SF-30%. If you consider just the right hand side of Fig 3, there seems to be a +ve to +ve relationship between higher SFs (SF10%, SF20% ,SF30% ) and LSS. The Pearson correlation between SF and LSS yields -0.421 (P < 0.05). 

Takeaway here is that GCT maybe a better predictor of LSS. Test Part B will prove if this is correct.

II. Results of Test Part B 

With some time given to practice, I was able to achieve a short and long GCT as follows :

Short GCT   226.6 ms
Long GCT   246 ms

The short GCT was obtained by conscious fore foot landing close to vertical line passing through COM and quick leg swings. The long GCT resulted from an exaggerated rolling motion of the feet, almost to the point of heel first first landing. Point of contact was farther away from the vertical line passing through COM. 

326 datapoints were consolidated from this test. 

Plots of data are presented below. 

Fig 4. Scatter plot showing % change LSS vs % change GCT during 
the variable GCT - fixed SF test.

Fig 4 :   Almost 96% of the variation in LSS was explained by variation in GCT. Further, the Pearson correlation between variable GCT and variable LSS was 0.98 (P < 0.05). The plot shows that variations in GCT in a variable GCT test - fixed SF test are more strongly related to LSS than in a variable GCT - variable SF test. This plot shows the LSS - SF relationship for both tests A and B making it clear the stronger regression in Test B (orange markers).

Takeaway is that lesser ground contact times are correlated with higher leg stiffnesses. (whether one causes is the other cannot be proved here. Also, that there maybe an injury limited stiffness value beyond which it is futile to stiffen the leg is also impossible to show here, since both tests rarely exceeded 12 kph. Theoretically, it seems possible that to generate high leg stiffnesses, the body may have to exaggerate dorsiflexion in the feet. But stiffness could also be a function of knee and hip torques so I refrain from digressing.)

III. Other Results 

A. Variation of Estimated Ground Reaction Force with Change in SF

Fig 5. Scatter plot showing peak nGRF expressed as N/kg with % variation in SF.

Fig 5 An expression for GRF was obtained from sinusoidal approximations given in Blum and verified with an older paper by Morin et. al. Using temporal variables measured by Stryd, the calculated GRF was then normalized by my body weight. This is shown in Fig.5.

The takeaway here is that nGRF decreases with increased SF, seemingly stabilizing between 
SF20%. and SF30%. Above the preferred step frequency, peak ground force decreases. This finding agrees with literature. The behavior at lower SFs could be a product of high COM elevation or aerial time. It tends to agree with certain reality that drastically reducing cadence at the same high running speed makes loud landing sounds in the gym. High force = high loading = high noise! This observation points to a flaw in the finding in Fig 1 that aerial time reduces with decreasing step frequency, infact I believe it should be the opposite. 

B. Variation of Rate of Force Production with Change in SF

Having seen in Part 1 that metabolic cost is directly linked to the inverse of GCT, it's interesting to inspect the behavior of this variable from the variable SF test data from Part A.

Fig 6. Scatter plot showing rate of force production vs % change in SF.

Fig 6 : There seems to be a distinct bow to the shape of this curve corresponding to the line of fcSF. Below and above this line at SF-30% ,SF-20% ,SF-10% ,SF10% ,SF20% & SF30% , inverse GCT shows increase. 

While data is limited to 626 samples and there is still some noisy data at the extreme low cadence, this graph generally agrees with literature. The surrogate data here suggests that there is a global minimum in metabolic cost at my freely chosen cadence of 177 SPM at 3.33 m/s pace. Intuitively, faster cadence means lesser GCTs. Muscle firing must be fast to produce ground force. Faster muscle firing has an oxygen penalty. Lesser cadence than preferred cadence at the same running speed also doesn't look optimal, as leg loading increases (Fig 5).

C. Vertical and Leg Spring Stiffness

One of Stryd's developers told me recently that they had not decided to release the math used to calculate leg spring stiffness (LSS) just yet. Curious as I am, I investigated how my own estimated values of ELS and Δy(COM) compare with Stryd's LSS and VO. 

EVS is the ratio of peak GRF and the vertical change in COM. ELS is the ratio of peak GRF and change in resting leg length. Both can be estimated from temporal variables and some high school math. 

C.1  Variation of Stryd Reported ELS vs % Change in SF

Fig 7. Scatter plot showing change in Stryd reported LSS and estimated VO with % change in SF.

Fig 7 : For lack of a metric to compare Stryd's LSS, I plotted it against the measured VO. The data suggests there's a curvilinear relation in both the variables with variable SF. LSS appears convex shaped against SF. The lowest VO corresponds to the highest SF's (and vice versa) whereas in leg spring stiffness, there appears to be a global minima between fcSF and SF10%.

The takeaway is that oscillation of a fixed point on my COM decreases with increasing step frequency. This agrees with the decreasing trend of FT and GCT at high SF's from Fig 1. Spring stiffness also increases with step frequencies. Both these findings tend to agree with literature.

C.2  Variation of estimated EVS vs % Change in SF

Fig 8. Scatter plot showing change in estimated EVS and estimated Δy(COM) with % change in SF.

Fig 8 : The change in estimated EVS is convex shaped with % change in SF whereas that of Δy(COM) is concave shaped. The data seems to suggest that global minima in EVS is somewhere between fcSF and SF10% and much higher in both directions from this point. Global minima in Δy(COM) is closer to the highest SFs and maxima is clustered around the lowest SFs. The magnitude of estimated EVS seems smaller than I expected compared against literature reported values.

The estimated value of vertical stiffness agrees with Stryd's LSS trend (Fig. 7) - both are convex shaped in SF and there appears to be a global minima between preferred cadence and SF10%

Variation of estimated ELS vs % Change in SF

Fig 9. Scatter plot showing change in estimated ELS and estimated ΔL with % change in SF. Note that ELS calculation corrects for the translation in point of foot contact, therefore it is a POFT corrected spring stiffness, yielding higher magnitudes than those calculated without POFT correction. 

The change in estimated ELS is convex shaped with % change in SF whereas that of ΔL is concave shaped. The data seems to suggest that global minima in ELS is somewhere around SF10% and much higher in both directions from this point. Global minima in ΔL is hard to tell, there seems to be two local minimas instead at the lowest and highest SFs each. The magnitude of estimated ELS also seems smaller than I expected compared against literature reported values. 

Comparing Figs. 8 and 9, the increase in effective vertical stiffness is more pronounced than effective leg spring stiffness at higher SFs. The magnitude of effective vertical stiffness is also higher than that of effective leg spring stiffness. This is in direct agreement with observed trends in literature (duh, most are using the same mathematics!)

Estimated ELS Compared to Stryd's Leg Spring Stiffness

Fig. 10.  Scatter and median value of bias magnitude between estimated Δy (COM) and VO reported by Stryd. Bias =  [Δy (COM) - VO ].

Fig 11. Scatter and median value of bias between estimated ELS and Stryd reported LSS.  
Bias =  ELS - LSS.

Figs. 10 & 11 : The last two plots show the difference in Stryd's metrics of VO and LSS vs estimated Δy(COM) and ELS respectively. 

The bias in estimated Δy(COM) and VO each is lowest around SF-30% and SF-20% and progressively increases with increasing SF. At SF10% and beyond, there is an average of 0.1m of difference.   

The bias (or delta) in estimated ELS is approximately 5 KN/m less than Stryd's LSS between SF-20%  and SF20%. At lower and higher SF's, the bias seems to get larger, with the highest bias of 10 kN/m at SF-30%

I'm not able to justify this difference, especially since ELS was estimated from Stryd's GCT and speed. The bias may result from a misunderstanding of how Stryd is calculating it's LSS. It could be possible that the GRFs measured by Stryd, if it does at all, are much higher than estimated GRFs from the spring mass model and they are using some inhouse fudge factors to agree with their test data. An explanation of these differences would be nice actually. However, what I have proven is that in the absense of any LSS data, just being able to use speed and a footpod that measures GCT will help you estimate ELS and EVS as surrogate data to measured LSS. I suppose trends are as instrucive as specific values.

In the next installment, I'll be continuing to stick with leg spring characteristics but with a changes in running speed. I will also hope to compare treadmill running with outdoor running to see what real world conditions impose on the metrics. 

Sunday, January 15, 2017

Actionable Intelligence for Running Part 1 : Ground Contact & Leg Spring Characteristics

This is what I hope will be a multiple part series on my lab efforts to study running characteristics and extract what I call 'actionable intelligence'. I hope it will be an enjoyable read. Readers are encouraged to submit findings of flaws to my email.  

The cost of transport is the average energy required for an animal to travel a unit distance and is composed of the mechanical power to :

1. Accelerate limbs with respect to center of mass during each stride. This is a change in kinetic energy

2. Raise and re-accelerate the center of mass against gravity during each stride. This is a change in potential energy.

3. Move against resistance in the environment by virtue of opposing effect of the medium, such as air resistance, friction or viscosity. This is resistive work. 

4. In the legs of larger animals, where both the front and hind legs can be used for braking and accelerating, muscles store elastic energy lost from potential and return some of this energy again on takeoff. It's a convenient way to preserve and recycle effort. So there is a change in elastic energy. 

The energy and power to supply the above are delivered by positive work (shortening) of some groups of muscles in conjunction with the negative work (lengthening) of some other groups of muscles.  The net energy equation then is :

Perspective is nice to have. Robert McNeil Alexander, the father of animal locomotion research, stated in the first chapter of his book on locomotion that it takes 15 Joules of energy for a human to increase the sum of kinetic and potential work in the course of a single step.

Running Metabolic Cost and Rate of Force Transfer

In a landmark paper by Kram and Taylor, metabolic rate of running was hypothesized to be proportional to the cost of supporting the animals' weight and the energy used by each gram of active muscle used in running. 

Observations established ground contact time, or the time available for each foot to generate this running force on the ground, to be inversely proportional to energy used by active muscle. Volume of active muscle was established to be proportional to body weight across a range of species and body sizes.

It became simple to describe metabolic rate of running. If Wb is body weight and ground contact time tc, the mass specific power can be established with a proportionality factor, c :

            EQ. 1

This spectacularly simple model explains 70-80% of the metabolic cost in both two legged and four legged vertebrates. 

A graph in that paper revealed the linear relationship of mass specific metabolic cost and ground contact time with speed. It showed the remarkable differences in running cost across a range of species of different sizes. Also shown is the largely speed independent co-efficient c, which is the ratio between mass specific metabolic cost and tc in EQ. 1. 

An important concept was the influence of body mass on energy cost mainly through rate of force transfer, or the reciprocal of ground contact time tc (1/tc).

If I were a rabbit escaping from a big dog straight out of hell, the rate of force generation I would need would be higher (greater 1/tc) than that of the dog because I need faster steps to cover the same ground (due to decreased step length), which involves lesser ground contact time, faster muscle firing and rapid rates of actin-myosin cross bridge cycling. 

Faster muscle firing means recruiting more energy intensive muscles. Conclusion : the metabolic rate expressed per kg of body weight would be higher for smaller animals than larger ones. This has been proven in animal locomotion research beyond any doubt.

Rate of force transfer, 1/tc, was shown to be a power function of running speed. But not on speed alone said a paper by Hoyt Those researchers displayed an intuitive plot of tc vs relative running speed, defined as running speed divided by leg length and explained that relative speed accounted for 97% of the variability in ground contact time tc. 

Running Economy and Ground Contact Time

Aerobic capacity, has been a subject of hot debate. Research found that athletes with similar maximal aerobic capacity values don't necessarily perform equally in distance running. Figure below shows two athletes with different running economies but similar VO2-dot-max values.

The opposite also seems to be true. Santos-Concejero showed that less efficient RE at velocity eliciting 10-km race pace in North African runners implies that their outstanding performance on the track at international athletic events appears not to be linked to running efficiency. It is possible that less efficient runners might be making up at world events by displaying greater VO2-dot-maxes.

The factors for describing running economy itself are complex. Investigating a few out of this bunch of variables give insight but leaving out the rest is problematic. This is why papers like that of Saunders make for good reading for a review of influencing factors of running economy.

Researchers have reported a 20-30% range in running economy among age, gender and performance matched groups of trained distance runners. Therefore, it was only natural that deeper interest would go into the structural characteristics of running motion to explain differences in metabolic demand and performance.

A paper by Weyand showed that in 36 healthy subjects, the rate of force transfer (inverse of ground contact time) was strongly related to maximal aerobic power across a range of steady speeds on a treadmill. 

Regression analysis showed 98.5% of the within-subject variation in mass specific maximal aerobic power was accounted for by 1/tc. 


Santos-Concejero, linked to earlier, also showed ground contact variables as having strongest correlations to the poor economy in North African runners vs European counterparts.

Similarly, Nummela showed that ground contact time was the only stride variable showing significant correlation with both running economy and maximal running speed in young well-trained endurance athletes, but wasn't sufficient by itself to explain running economy in the most economical runners.

Below is a graph showing the decrease in ground contact times (black dots) and aerial times (white dots) with increase in speed in Nummela's study.

Again, correlation is not causation. But short contact time, according to the Nummela paper, seemed logically beneficial for both running economy and maximal running speed since the critical point in maximal sprint running and economical running is the speed lost during the braking phase of a run.

Leg Spring Characteristics

A staple in engineering science, the mass spring elastic model came to fantastic help to analyze running motion. Several researchers have used the spring loaded inverted pendulum (SLIP) model (without damping) to accurately describe locomotion, whether that be trotting, hopping, or running, in a range of body sizes across 3 orders of magnitude. 

It turns out that whole-body center of-mass motion during running is similar to a pogo-stick (spring-mass) model where the leg can be represented effectively as a spring. So forget the complex neuromuscular stuff going on within the body. You can describe leg physiology quite nearly as the real thing with a linear spring model. 

A direct result of the linear spring is that the calculated work rate in the spring must equal the work rate in the center of mass.  If the leg cannot be assumed as a linear spring, this equivalence is flawed. 

The mass specific work rate of the leg spring, Pspring or mechanical power output, is established as reciprocal of leg spring stiffness multiplied by the product of the square of spring force (F), step frequency (f) and body mass M. 

In biomechanical research, there appears to be several calculation techniques to estimate leg spring stiffness. To derive each of these secondary factors would be pedantic, but one paper by Blum et. al nicely summed up the "difference in the math"  : 

The authors suggest that independent of low and high speed running regimes, method E is a simple and robust technique to estimate leg spring stiffness based on the spring model. In the low speed regime, a simpler calculation method such as A was found to suffice.

Another point of contention arises with the assumption in the SLIP model that the foot's point of the contact with the ground does not move. In reality, this is not true. When humans walk or run they tend to make initial ground contact with the heel or the middle part of the foot, and then to roll the foot forwards and lift off from the toe. This moves the point of contact. The SLIP model was modified with the point of force translation model (POFT) to portray this reality, first in this paper by Bullimore and associates. 

The difference in the two representations are shown below, showing how choice of methodology can affect the calculated value of leg spring stiffness :


L = initial leg length
ΔL = leg length variation (compression) during contact in the classical SLIP model
ΔL' = leg length variation (compression) during contact in the POFT model
Δy = vertical downward peak displacement of the center of mass during contact
tc = ground contact time
Fmax = peak ground reaction force
d = point of force translation distance
v = forward running velocity

One words of caution. Different calculation techniques to estimate the compression of the leg while deriving leg spring stiffness can influence overall findings, sometimes even misguide. 

Papers written in the early 90's like this were stating that leg spring stiffness was observed to be constant across forward running speeds. Later, some papers like this one stated that leg spring stiffness could change dramatically with different stride frequencies at the same speed although leg spring stiffness was constant across running speeds. Most of these papers were using a calculation method advanced by McMahon and Cheng

Arampatzis corrected that leg spring stiffness is influenced by running velocity and influenced by the increase in knee spring stiffness rather than ankle spring stiffness. They also stated that older methods such as those of McMahon and Cheng overestimated the change in leg length in the spring mass model causing leg spring stiffness to be constant across velocities.

There is a third spring analysis method, called the actuated SLIP model with actuation. This method assumes damping and actuation in the leg spring to represent the combined effects of both hip and ankle torque during locomotion. In contrast to the undamped linear spring model of classical SLIP, this method is a non-linear model of running mechanics :  

This extension of the SLIP model with active torque and damping is the simplest established model of legged locomotion that is capable of predicting the mechanical cost of transport. The reason is that by including damping in the leg, the analyst is able to account for the concept of stability in locomotive movement and isolate the non-zero energetic cost to ensure stable motion.

A unique and recent paper written by two mechanical engineers from Purdue, which used the SLIP model with actuation, suggests that when non-dimensionalized for body weight and resting leg length, an optimum "relative leg spring stiffness" was selected by a running organism. This optimum corresponded to the minimize cost of transportation to the organism.

Interestingly, Shen and Siepel found that with the SLIP actuation model, experimental values for leg stiffness in humans matched closely with the simulated relative leg stiffness corresponding to energy minima.

That there could be one, or very narrow range of operational values for leg stiffness, over a range of speeds and leg landing angles is very interesting.  This speed-independent biological preference for optimum leg stiffness also means that all animals prefer to stay somewhere in a narrow range of between 7 and 27 in relative leg stiffness. 

This is a new paper, therefore citations to the work are relatively low in number.  The concept of an energy minima limited leg stiffness makes complete sense and I'm glad that engineers, using analysis tools in dynamical control systems, are lending a multi-disciplinary hand to the whole running biomechanics investigation.  

I effectively stop here in reading papers. The total volume of research in running biomechanics span decades and it is useful not to read every one but establish general principles that research provides.

1) First, that 1/tc, which is the rate of force transfer, is an important running parameter, which explains 70-90% of the energy cost is bipedal organisms.

2) Several factors affect running economy but generally it's been found that ground contact variables, such as tc, have stronger correlations with running economy and maximal speeds.

3) Simplistic linear relationship cannot explain everything going into the energy cost of running. For example, rate of force transfer, is not just related to speed but also factors like leg length.

4) The spring mass model is remarkably powerful to understand running dynamics, inspite of the neuromuscular complexity in the leg. But using incorrect equations to calculate leg spring characteristics result in misguided conclusions about it's relationship with speed.

5)  Leg spring stiffness may or may not be trainable, although there is evidence that this parameter is a strong function of ground contact time.  New research provides evidence of an energy minima limited leg stiffness value when normalized by body weight and resting leg length. 

6) Research is in flux and what was believed 3 decades ago might not be true anymore. Interdisciplinary facets to investigations lend additional insight into the science of running, removing age old myths and misconceptions.

It is important that I get to my own mini-investigations of what ground contact time and leg spring stiffness mean to me as a runner, which will be the subject of part 2.