How is power on PM5 calculated?

[Nomath]

...Then use HR, if that doesn't cover all work done, what can?

A HR monitor is very useful, but it is not an ergometer.

[jamesg]

A HR monitor is very useful, but it is not an ergometer.

Incorrect, it is, and not bad at least in the aerobic range.

You only have to find the ratio between Watts and HR range in use, using a C2 ergometer.

[JaapvanE]

You only have to find the ratio between Watts and HR range in use, using a C2 ergometer.

That is far from a constant ratio as technique, fitness, machine settings and environmental conditions all influence this ratio.

[jamesg]

Not from my observations. What have you seen? And why do do many people put their trust in HR as a proxy for Power?

[Carl Watts]

Not from my observations. What have you seen? And why do do many people put their trust in HR as a proxy for Power?

I'm really surprised when looking at peoples results or just watching on Zwift the number of people not using a HR monitor.

Its a vital piece of training kit and I never get on the rower or bike without in on.

I guess some people just see it as an extra expense and a hassle changing batteries.

Pretty sure the latest change in Zwift corelates your power to HR at the bottom of the screen as a graph.

[JaapvanE]

Not from my observations. What have you seen? And why do do many people put their trust in HR as a proxy for Power?

One shouldn't do that, as it is a bad indicator.

When I have a heavy diner before rowing, my HR is 10 beats higher than normal. When I had fizzy drinks before rowing, my HR is all over the place. When room temperature, humidity or CO2 levels are too high, HR goes way up. When drag changes, so does HR. When you switch from a high SPM/light stroke to a low SPM/heavy stroke, results are incomparable. HR is a generally decent stress indicator (although HR variability does a much better job), but it doesn't discriminate which stressor. So environmental conditions will play havoc with these relations.

My practical experience which I actually measured directly: my Wahoo Headwind fan saves me 10 to 15 BPM regardless of pace. Just because I'm better cooled, even in winter. That is quite a difference, which has nothing to with power generated, but all with the way more efficient heat removal from the body.

Garmin (or more specifically FirstBeat) actually correct fitness measurements for environmental conditions for this specific reason: when summer is arriving, people's general HR goes up due to the stress heat causes.

I mean in a very controlled setting, it might be a half-decent proxy. I have used it over the course of a year, where random factors are expected to level out, to calibrate my NordicTrack to my gym's C2. Still I was off by 20 Watt (i.e. 10%) when I could really compare times in identical circumstances (i.e. really side-by-side).

And we actually train with the explicit aim to change that ratio. When I look at VO2Max, which is a pretty accepted metric for cardio-fitness, much of the measurement protocols are actually based on the relation between power produced and HR. The global idea is that cardiovascular fitness increases when you are capable of pulling more watts for the same HR. Regardless what people think of VO2Max, I think we can all agree that somebodies fitness has increased if he/she is capable of pulling the same times with a much lower heartrate.

I know that I can row the same distance at the same pace, but instead of last years Zone 3, it is now a Zone 2 activity. In part because of more efficient technique, in part due to an increased general fitness. When I'll go past 60, it will probably drop again, as power is degrading and thus generating the same power will become harder and harder.

[JaapvanE]

I'm really surprised when looking at peoples results or just watching on Zwift the number of people not using a HR monitor.

Its a vital piece of training kit and I never get on the rower or bike without in on.

Agreed. I recently discovered breathing rythem as an additional good indicator to watch in rowing. My lactate transition point can be fairly decently estimated by seeing my breathing rythem. When it is nearly identical to my stroke rate, I'm in HR zone 2, when it switches to double my stroke rate, I'm in Zone 3.

[jamesg]

When I have a heavy dinner before rowing, my HR is 10 beats higher than normal. When I had fizzy drinks before rowing, my HR is all over the place. When room temperature, humidity or CO2 levels are too high, HR goes way up. When drag changes, so does HR.

Under such conditions, worrying whether the C2 machine is an accurate ergometer or not might seem somewhat absurd.

When you switch from a high SPM/light stroke to a low SPM/heavy stroke, results are incomparable. When drag changes, so does HR.

You do that?

[JaapvanE]

When I have a heavy dinner before rowing, my HR is 10 beats higher than normal. When I had fizzy drinks before rowing, my HR is all over the place. When room temperature, humidity or CO2 levels are too high, HR goes way up. When drag changes, so does HR.

Under such conditions, worrying whether the C2 machine is an accurate ergometer or not might seem somewhat absurd.

Why? An Ergometer should measure what it can, as accururate as it can. And the C2 PM5 just does that.

Humans are not robots. Human physiology is a hugely complex thing. Considering HR to be a proxy for externally delivered power is flawed at best as way too many variables come into it.

As said: environmental conditions can have huge effects. Even stress from work causes HR to change. Time of day varies as people often don't have the luxury to train when it is the absolute right time to do so as work or family takes preference. That in itself (aside any fatigue that will come with training later in the day) will cause variation.

As an illustration: I usually walk a 5K with my wife every day. Pace is nearly identical, but average HR last monday was 75, last friday it was 88, and I recall a 95ish a couple of weeks ago. So quite some variation, just in a simple 5K walk with identical pace on an identical course I've been doing for years now.

And this isn't odd. This is the basis of many metrics used by Garmin, Apple, etc.. That is why Garmin even mentions it corrects the results for non-optimal environmental condtions.

If I recall correctly, Carl records temperature and humidity with his rows. I also include CO2, as the wind direction determines how well my room is ventilated, and thus how much fresh air gets in. When there is little to no wind, CO2 levels rise quite dramatically during a row, affecting HR. And it should: the higher CO2 level causes my body to work harder to get O2 out of the air.

When you switch from a high SPM/light stroke to a low SPM/heavy stroke, results are incomparable. When drag changes, so does HR.

You do that?

I'm building up after an injury, so yes, I do that on an almost daily basis. Given the effects, I might even keep it: a higher drag on the short pieces (everything under 10K) fits me better as I can generate more power. On the longer stuff I like the smoother stroke, as the power endurance isn't there (yet?).

[Nomath]

Back to the question : How is power on PM5 calculated?

We don't know the precise answer because the software in the PM5 has never been disclosed. It is speculated in the above that C2 uses the approximation :
P = k * ωm³
ωm is the mean angular velocity during the whole stroke, i.e. drive + recovery ; k is the drag factor in [kg m²].

I showed above that this approximation gets remarkably close to the true value in a steady state, when successive stroke have equal power. It is easy to see that when successive strokes get stronger, the flywheel accelerates from stroke to stroke and the formula underestimates the true power input. On the other hand , if successive strokes get weaker, the formula overestimates the power of the stroke, because part of the drag power is drained from the kinetic energy of the flywheel.

A more accurate formula is to take the loss or gain in kinetic energy of the flywheel from stroke to stroke into account :
P = k * ωm³ + ½ * J * (ωm² - ωml²)/∆t
J is the moment of inertia of the flywheel [kg m²], ωml is the mean angular velocity of the previous stroke and ∆t is the stroke time.
I don't know whether C2 uses this formula or not, but I expect that it is more accurate than the first formula in an unsteady stroke pattern.

To demonstrate the gain in accuracy due to the extended formula, I did a computer simulation of 7 consecutive strokes using the Human-B force profile published by the Ulm University group. The different force profiles shown in the figure were measured for elite rowers during steady state rowing. The horizontal axis is the travel distance of the handle from the catch point.



To use a more 'average-Joe' force profile, I scaled the maximum force to 600N and the travel distance to 1.50 meter but kept the shape identical.
Note that the area under the curve equals the energy input (work) to the flywheel by the rower. With the reduced force and distance the work equals 505.7 Joule.
The drag factor was set to 120 (i.e. k = 0.000120 kg m²)
The speed and duration of the drive depend only on the angular velocity at the catch, the force profile and the drag factor. The duration of the recovery in stroke #1-3 was set to 1 sec. For strokes #4-7 the total duration of the stroke was set to 2.4 sec, hence 25 spm.
The figure below shows the angular velocity as a function of time. The vertical lines are at the start of the drive. The blue line is the average angular velocity in each stroke.



Because the force profile is rather odd for a start at zero flywheel , the first drive takes nearly 10 seconds! The second drive is much quicker : about 1 second. Later drives are finished in less than 0.9 sec. Because the recovery time in stroke #3 was set to 1.0 sec and in strokes #4-7 was about 1.5 sec, the average flywheel speed is seen to drop.

The work done in each stroke is constant : 505.7 J
Because of the different stroke times, the exact average power in strokes #1-7 is :
47W - 247W - 266W - 211W - 211W - 211W - 211W
Using the formula P = k * ωm³ the calculated power in strokes #1-7 is :
49 W - 152W - 213W - 233W - 217W - 212W - 212W
Using the extended formula, the calculated power in strokes #1-7 is :
74W - 303W - 282W - 251W - 202W - 207W - 211W

To our surprise, at least to my own surprise, the extended formula does not lead to a better estimate of the power in the stroke!
How come? Physically the extended formula appears more accurate. It took me some time to figure this out. Eventually, I realised that the definition of a stroke as Drive + Recovery is wrong. The proper definition is Recovery + Drive ! Why? Because what happens in the drive is determined by the angular velocity at the catch, the force profile and the drag factor. The angular velocity at the catch is determined by the prior recovery. Consequently the mean angular velocity should also be taken over the Recovery + Drive. Compare the graphs above and below (below, vertical ines are at the start of the recovery).



Using the revised definition of a stroke, the exact power in strokes #1-7 is :
52W - 247W - 266W - 274W - 208W - 210W - 210W
Using the formula P = k * ωm³ the calculated power in strokes #1-7 is :
24W - 108W - 187W - 233W - 222W - 214W - 211W
Using the extended formula the calculated power in strokes #1-7 is :
41W - 252W - 295W - 290W - 213W - 206W - 209W

Clearly, the results of the extended formula are much more accurate.
Computationally, the additional term only requires that mean angular velocity of the previous stroke has been saved and the stroke time is measured. This is quite simple and robust.

Again, it is not sure how the PM calculates power. But we know that on the PM display the power is updated after the drive, not after the recovery. This agrees with the above reasoning.

[JaapvanE]

A more accurate formula is to take the loss or gain in kinetic energy of the flywheel from stroke to stroke into account :
P = k * ωm³ + ½ * J * (ωm² - ωml²)/∆t
J is the moment of inertia of the flywheel [kg m²], ωml is the mean angular velocity of the previous stroke and ∆t is the stroke time.
I don't know whether C2 uses this formula or not, but I expect that it is more accurate than the first formula in an unsteady stroke pattern.

That is an interesting formula! I'll put that to the test very soon.

To our surprise, at least to my own surprise, the extended formula does not lead to a better estimate of the power in the stroke!

That is extremely surprising indeed. I would have expected better results as well.

How come? Physically the extended formula appears more accurate. It took me some time to figure this out. Eventually, I realised that the definition of a stroke as Drive + Recovery is wrong. The proper definition is Recovery + Drive ! Why? Because what happens in the drive is determined by the angular velocity at the catch, the force profile and the drag factor. The angular velocity at the catch is determined by the prior recovery. Consequently the mean angular velocity should also be taken over the Recovery + Drive. Compare the graphs above and below (below, vertical ines are at the start of the recovery).

Interesting. With ORM we already calculate power every half stroke (i.e. end of recovery or drive). I'l test this side by side.

Clearly, the results of the extended formula are much more accurate.

Unexpected, surprising, but what the data tells us. I'll do some experiments as well.

Computationally, the additional term only requires that mean angular velocity of the previous stroke has been saved and the stroke time is measured. This is quite simple and robust.

I agree with you on this one. I assumed you could only apply this with instantaneous angular velocities and accelerations. So this is a very interesting avenue to persue indeed.

Again, it is not sure how the PM calculates power. But we know that on the PM display the power is updated after the drive, not after the recovery. This agrees with the above reasoning.

Look at the writings of Prof. van Holst (you may need to use a web archive to get them, this seems to be a good copy, otherwise I can send you a copy). He discussed a similar possibility about power calculation and stroke definition.