STEP 1 : The X, Y and Z velocity functions are an integral of the ratio of the respective ground reaction forces and mass, or in other words the axis specific accelerations. After integration is performed, a constant has to be added to the right side which maybe different for level running and gradient running (Cavagna constant).
STEP 2 : Since velocity is a vector, the resultant running velocity is calculated by the sum of the squares of the 3D velocities.
STEP 3 : Kinetic energy as a function of time is half of mass times the resultant velocity squared.
STEP 4 : Vertical displacement of center of mass is the integral of vertical velocity with respect to time. After integration is performed, a constant has to be added on the right side (Cavagna constant).
STEP 5 : The potential energy as a function of time is mass times gravitational acceleration times the vertical displacement of center of mass.
STEP 6 : The total energy of the center of mass as a function of time is a summation of the potential and kinetic energies. This will fluctuation function in time, with a minimum occuring at the middle of stance phase. For level running, the curve is symmetric. For gradient running, the curve is lopsided to the right side for uphills and to the left side for downhills.
STEP 7 : External Power is the time derivative of the summation of changes in potential and kinetic energy functions.
PART B : Stryd Whitepaper Review
I. Review of Stryd's Vertical Force-Time Curve
Vertical power is the rate of change in potential energy. Multiple studies have shown that this component constitutes the majority share in the metabolic cost of normal running.
Since the Stryd footpod has no means to measure force directly, in order to get the correct estimate of force, the IMU must capture the time course of instantaneous vertical acceleration profile which is proportional to the force signature.
Continuing on, in order to get an accurate estimate of changes in potential work, the footpod IMU must capture time course of vertical displacement of center of mass.
An image showing the first of these elements is in Figure 2.
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| Figure 3 : Modeled vs actual vertical force-time signatures in a rear-footed runner for an unspecified running speed. Base image courtesy of Stryd. Markups by the me. |
It is not known what speed the runner was running at but looking at the force signature, it is clear that it was a rear foot runner. The model estimation and force plate data nearly match, particularly when you look at the highest peak vertical force which is important in derivative calculations of impulse and leg spring stiffness.
Some key observations as far as
vertical force-time (Figure 1) is concerned :
1) Misrepresentation of first footstrike peak : The Stryd models a nice rounded single peak force when there were actually two. This is significant because the footpod thinks that the footstrike is fore-footed when in reality it's rear-footed. The implications are :
a) The Stryd model neglects first impact peak. If the first rear-foot strike peak is completely neglected, that initial impact peak of landing will be distributed over an inaccurate and wider period of delta time which will suggest that there is no shock loading, when in reality there is.
b) Stryd's force profile estimation for a forward moving (or backward moving) fore-footed runner might be alright but it might be grossly over-looking rear footed mechanics by missing the first impact peak altogether.
c) If a runner changes gait mechanics on the fly due to the effect of different running surfaces or due to fatigue in a long race, this 'tuning' may not be captured properly by the model.
d) Gradients may substantially influence these errors. To get an idea of grade influences on the vertical force-time profile from empirical studies, please see Appendix image A4.
2) Mismatch of contact time : Although the actual and estimated signals are close, Stryd under-estimates initial footstrike and over-estimates the actual takeoff point with respect to time (Figure 2).
This has some implications, namely :
a) The ground contact time (GCT) is over-estimated. From the image, I estimate atleast 15-20 milliseconds greater than the force plate.
b) The stance-averaged vertical ground reaction force during GCT does not match 1-to-1 with the actual force plate data.
c) As a consequence of b), the Stryd estimated ground reaction impulse given by the product of force and GCT is different to the actual impulse.
d) Any calculation of aerial times, step lengths and step times using GCT will propagate the error through.
e) Since leg spring stiffness (LSS) is driven by GCT and involves a duty factor calculation treated with the maximum vertical ground reaction force, errors propogate into the LSS model as well.
f) Since the tests were carried out in the laboratory, the effect of gradients and different speeds on the error in GCT remains unknown.
3) Effect of Running Speed on Force-Time Curve Not Discussed : Stryd does not show the effect of running speed on the goodness-of-fit for vertical force curve.
That this important element is missing in the whitepaper prevents a discussion on the influence of variations among broader running speeds and broader gait mechanics. Within literature, researchers have found that simplistic vertical force-time curve models derived from spring mass models lose their goodness-of-fit as the running speed increases due to the presence of high frequency components from the acceleration of the lower limb.
This is where I'd exercise some caution. With the current state-of-the-art, I wonder if IMU's may still not be practical for application to short-distance, high speed track racing.
4) Impact of Shoe Type to Force-Time Curve Not Discussed : The impact of variations in footwear to the measured parameters is unknown. This is also an influencer of vertical ground reaction force profiles.
5) Impact of Treadmill Slope to Force-Time Curve Not Discussed : The whitepaper lacks a review of the force-time profile accuracy under the influence of slope (see Appendix image A4). Since step period generally decreases as slope increases and increases as slope decreases, what influence high running speeds have on the model fit when the slopes are greater than 7 degrees inclination is something important to document.
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| Figure 4 : Plot showing goodness-of-fit of modeled GCT to force plate measurements. The average error is stated to be 2.83%. The number of runners, running speeds, shoes worn, footstrike mechanics and slope on the treadmill are all unknown which raises the question of how the error varies as a function of each. Image courtesy of Stryd. |
The second element in estimating potential energy changes is
vertical oscillation, shown in Figure 3.
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| Figure 5: Estmated vs actual vertical oscillation across a wide range of runners. Sample size is not disclosed. Experimental method and running speeds to generate the plot is unknown. Stated average error with respect to force plate = 3.18%. Base image courtesy of Stryd. Markups by me. |
Some key observations as far as vertical oscillation is concerned :
1) A close fit : A 3.18% average error in vertical oscillation has been shown which is quite good. It is desirable to understand the experimental method, equipment used and the sample size of the runners to put this into context.
2) Error Propagation : I understand that a small error propagates into the Form Power calculation due to vertical oscillation error.
3) The effect of gait parameters on this variation is not documented. The estimation error with respect to gait parameters such as velocity and step duration should be additionally plotted, for example, in the form of a Bland-Altman plot.
4) Explanation of influencing factors behind error : I'm very interested to know if 3% average error is the best achievable given current level of technology. It is desirable to get some sort of explanation to the influencers of this error. Do random influences play into this?
II. Review of Stryd's Horiztonal Force Model
Horizontal power is a function of the rate of changes in kinetic energy. Studies have suggested that the horizontal work, particularly that component of generating horizontal propulsive force, constitutes more than one-third of the total cost of steady speed running.
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| Figure 6 : Stryd modeled forward speed change compared to force plate measures. Indicated accuracy = 95%. Grade of running surface unknown, but presumably level. Image courtesy of Stryd. |
A review shows the following :
1) Accuracy is decent : Like the vertical force-time profile, the horizontal force-time profile is presumably derived from the horizontal acceleration signature from the IMU. Stryd states that it captures the speed attenuation during the contact phase with 95% accuracy. This is not bad when considering that this anterio-posterior dynamics is a difficult one to capture.
2) Error Propagation : A 5% error in estimation of kinetic energy change propagates a proportional error into the work and power calculations.
3) Influence of speed and slope not discussed : Like in the case of the modeled vertical force-time profile, the influence of different speeds and slopes on the goodness-of-fit are not specifically mentioned. This is important to document in order to compare how the fit varies under different situations. See Appendix image A4 for grade influences on horizontal force-time curve curves.
III. Basis of concentric and eccentric work scaling factors not discussed : A technical basis for the scaling factors employed by the Stryd model to account for the dominance of concentric or eccentric work during gradient running is desirable (see Part A.II and Figure 1).
This correction was made as a firmware update in early 2017.
While I understand this is part of the secret sauce, some pedantic questions are necessary to be asked if we want to remain true to the estimation of a power :
A) Is the scaling used related more to vertical and horiontal ground forces or is it calibrated with metabolic costs? What was the validation study behind these and will those findings translate well for the general public?
B) Is the power scaling a continuous linear decrease for downhill grades and a continuous linear increase for uphill grades?
Let's suppose scaling is calibrated against metabolic costs. If power follows a strictly linear decrease for steep downhills while the metabolic cost decrease sharply, this will suggest that the metabolic efficiency increases. Conversely, if power follows a strictly linear increase for uphills while the metabolic cost follows a curvilinear relation to grade , that might suggest that metabolic efficiency becomes progressively worse. There are individual variations playing into metabolic cost dynamics on grade.
For downhills, a strict metabolic cost decrease may not even hold. For example, it has been documented that beyond a grade of -9 degrees, the metabolic rate actually increases, presumably from the high eccentric cost to maintain balance of center of mass. In other words, there is an optimum downhill angle beyond which metabolic cost increases.
How that scaling curve has been calibrated is of much interest to me, and I assume, to other scientifically minded runners.
C) Some reports indicate runners losing RPE-power correlation for gradient running. This begs the question whether the scaling factors should be something that is best left for the runner to tune and calibrate through the settings instead of being driven down from Stryd.
VI. A Review of Stryd Statement on Correlation with VO2
Stryd states in the whitepaper that : "The external mechanical power reported by Stryd has a well established relationship with metabolic expenditure based on testing conducted by Stryd and other third party research teams. "
Now I have done this too in the
past during a VO2max test, plotting the relationship of VO2 to W/kg. This can be done by absolutely anyone.
I consider the statement regarding "well-established relationship" to be a sleight of hand. In other words, if I take a powermeter that is algorithmically modeled in such away that external power is linearly proportional to running speed (atleast on flat terrain), ofcourse a VO2 test is going to show that ave VO2 at each speed is proportional to external power! This relationship has been "pegged" from the beginning due to speed being an input in the model. There is no unique science in this.
The linear relationship of VO2 to power is encouraging in as far as it only tells us that model algorithm involving speed works.
The next question would be : Can you use a running power meter be used to predict running economy? This carries a risk of mis-estimation because we do not know how transferable such simple relations are going from indoors to outdoors.
The estimation gets worse when it is a formula derived from a book which based it on data from a limited sample of runners that you weren't a part of. I do not believe you can estimate metabolic cost this way with any reliable degree of accuracy just like HR or HR based formulas cannot estimate caloric expenditure with any reliable degree of accuracy.
Indeed, a study from the University of Guelph and presented at the recent Canadian Society for Exercise Physiology (CSEP) annual meeting in Winnipe challenged the idea.
The researchers found a significant difference in running economy between treadmill and track running for 11 experienced elite runners as measured by standard metabolic measurements. But in the same study, the Stryd power meter and formulaic implementations of economy couldn't pick up any difference between the two surfaces.
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| Figure 7 : Plot showing VO2 as a function of specific external power reported by Stryd. Since power is linearly proportional to speed, such a relationship is already mostly pegged by design of the algorithm. |
OTHER ASPECTS :
V. Leg Spring Stiffness : Information Only Or Actually Used?
For level running, we understand from scientific literature that the storage and recoil of energy in the lower limbs restores an appreciable amount of energy into the positive work phase. Thus, the mechanical efficiency of running maybe greater than 30% depending on the skill and mechanics of the runner. This efficiency of running has a documented linear relationship to running speed.
Stryd calculates leg spring stiffness (LSS) using published models derived from the mass-spring paradigm. But it is unclear how Stryd's model employs the stiffness into the mechanical work calculations to account for a "savings" in concentric work requirements. In contrast, the GOVSS model appreciates there maybe savings from efficiency increases as a function of speed and corrects the power demand depending on running speed.
So the question is whether leg spring stiffness is simply a metric displayed for information only or whether it is actually used in the model in a fashion as described in the para above? If it is not used, then someone can question the actual value of this. The LSS metric remains one of the most confusing metrics from a trainability standpoint.
VI. Apparent Mechanical Efficiency of Running = 25%
The apparent mechanical efficiency of running is defined as :
Meff, a = ratio of external power (Pext) and metabolic rate (Pmet). Meff, a = Pext/Pmet.
The gross mechanical efficiency of running is differentiated as :
Meff,g = ratio of total power (Ptot) and metabolic rate (Pmet). Meff,g = Ptot/Pmet.
In the context of a Stryd powermeter, we should be concerned about the apparent mechanical efficiency.
As alluded to in section V, human running involves energy storage and recoil going from the negative to positive phase. Several researchers have found that upto 40-50% of the energy stored during the eccentric phase can be returned to the concentric phase within the short time span of ground contact time for which those muscles remain in a loaded state.
Maximum possible elastic energy storage is defined in some papers (such as those written by Kram et.al) to be the difference between initial and minimum external energy of center of mass during the stance phase. Energy return is defined in the same papers to be the difference between the ending and minimum external energy of center of mass during the stance phase. The
maximum possible energy storage and recovery is then taken as the smaller of these two values.
To me, the way the efficiency is defined and what it takes into account (or what it doesn't) explains a lot of the differences in calculated efficiency between different running power models now arriving in the market. I suspect that an apparent mechanical efficiency value of 25% is artificially low, atleast for running on level and shallow slopes, if it didn't take into account energy storage and recovery mechanics between the negative and positive phases of running.
VII. Effect of Wind on Stryd's Performance
The Stryd powermeter has no way to account for wind effects in the power calculation. In this respect, it will under-report power by a factor proportional to the correct relative velocity cubed. By "correct", I mean that the wind measured has to be applied at the height of the runner, and not what is reported from a 10m or 30m wind tower.
That said, one of the "buggy" issues, as has been reported by several people both on the
Stryd support forums and the
Running Power Google Groups, is the sensitivity of power to sudden wind gusts. Reports indicate that gusts cause unsteady spikes in power for some and dips in the reported power for others, which is physically incorrect if you were trying to maintain speed in the face of a headwind.
The root technical fault maybe with the barometer, which thinks that the a sudden pressure front is a change in pressure altitude. How that information is relayed through sensor fusion and into the Stryd algorithm to mess up the reported power is a mystery to me. Judging by the forums, even Stryd's engineers have a challenge grappling with this issue.
The fact that they may need several consistently gusty days outside to test what's wrong might mean the delay of a corrective action for users.
VIII. Conclusions
Stryd's description of the external running power model and a comparison of modeled variables against force plate data has been long in the making. It is appreciated but delivered a bit late.
From a brief reading, I assess that they employ the general EESA approach to external power with some "in-house" tweaking for uphills and downhills to account for net energy addition or dissipation.
Stryd is thinking several things, some unique, some literature driven, about the kinematics of running. I give them credit for that. However, it does not stop the questions about how the model employed will validate for a large number of runners in actual usage. This same question also goes for the GOVSS run power model.
The effect of running speed, footstrike variations and slopes on those errors were largely missing from the whitepaper. This was the most important aspect I would have liked to see. This unfortunately prevents an assessment of how closely IMUs can correctly decifer footstrike patterns across a broad range of runners, running speeds and terrain.
Though the stated errors in key variables and things like force-time curves are small, those errors propagate into the calculations of derived metrics. Users must be aware of this when trying to introduce running interventions to effect a change in some of these metrics.
It is hoped that this technical review will encourage them to release another round of whitepapers so we can understand that aspect. An independant scientific review from other laboratories is also desirable in order to establish the degree of reproducibility in these numbers.
With Stryd and Runscribe having published their running models, the lights fall onto Garmin. With a far greater user base, they should find impetus to publish their running power framework soon or risk a lukewarm interest from the market.
In the next post, I'll explore how errors in estimated ground reaction forces translate into errors in the external power calculations from the EESA method. Stay tuned...
APPENDIX
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| Figure A1 : An illustration of vertical ground reaction force-time curve along the gait cycle. Courtesy Weyand et.al (2010). |
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Figure A2 : An illustration of the horizontal ground reaction force-time curve (lower plot). Courtesy Farley & Ferris.
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Figure A3 : Specific vertical force-time profiles for unshod rearfoot-striker, shod
rear foot-striker and a barefoot forefoot-striker at 3.5 m/s running speed. Courtesy Liebermann et.al (2010).
Figure A4 : Vertical and horizontal force-time curves for a 73kg subject running at 3 m/s over the indicated grades. For the vertical force profile, the first impact peak substantially increases as grade plummets. On the uphills, the second peak substantially increases to the point where at +9 degrees, the slope is rounded. Peak vertical forces decrease as grade steepens. For the horizontal force profile, the negative part of the S curve substantially increases as grade plummets while the curve more or less assumes a half sinusoid. On the uphills, the positive part of the S curve substantially increases as the grade increases while the curve as a whole more or less assumes a half sinusoid. Courtesy Gotschall et.al (2005).
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Figure A5 : Submaximal VO2 is linearly related to speed. Courtesy Kram et.al.
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