Monday, April 26, 2010

bike ban on East Road in the Headlands

The forbidden way... click for larger map.

"No Bikes" is all the sign at the entrance to East Road says. The old sign, which said "due to gravel" along with how vehicles and pedestrians were still allowed, was gone. I guess they decided it was time to get serious.

That sealed the deal. I turned onto East Road, rather than continue on the more direct route up Alexander.

It's all completely ridiculous. There's a few trenches, marked with ominous "open trench" signs, which are filled with dirt and covered with gravel. Sanchez Road between Market and Duboce in the City is far worse. Yet for some reason cyclists, the majority of the traffic mountain bikes designed for riding up fire roads and single-track, is forbidden. Oh, my -- a cyclist might hit the gravel and crash! But so might a motorcyclist, or for that matter any vehicle. During the week the road is completely closed to all traffic, but on weekends, cars and pedestrians are allowed. Only cyclists are denied access.

In California, cyclists may only be banned from freeways, tunnels, bridges, and expressways (a recent addition to the list). I suppose temporary emergency situations which might be particularly hazardous to cyclists, such as a power line downed in a storm, might also apply. But East Road is different: it's national park land, part of the Golden Gate National Park. The Federal government is far less pro-cyclist than California, at least in talking the talk. The California Bike Coalition and the San Francisco Bike Coalition have each done fantastic work towards promoting the rights of cyclists to use the roads. The League of American Bicyclists far less successful, in my humble view. In fact, only recently has the federal government made it a policy statement that cyclist access be a priority in transportation projects. And for this, Transportation Secretary La Hood has taken a lot of flak from the radical motorheads, who very much like the monopoly their carbon-spewing behemoths of steel and plastic have on the federal transportation infrastructure.

The whole Headlands project, of which the East Road trenches are a part, promises to improve the conditions for cyclists there. But it's all porkified hogwash. Conditions for cyclists in the headlands now are almost idyllic, all except for the tunnel on Bunker Road, which at least has a trigger for cyclists to illuminate a warning sign of their presence in the tunnel. No -- the purpose of the project is to facilitate car access: to "handle the rigors of 21st century tourism." Whoa -- I thought the "drive the SUV up the hill, take a photo, and drive back down" paradigm was the folly of late-20th century tourism, hopefully not the 21st. This century, we've realized we've got to be smarter managing the movement of people and objects: hauling 6000 lb of mass-produced, over-marketed junk up a 1000 foot climb to carry a 150 lb tourist (okay, I'm optimistic: most are a lot more than 150 lb...) to a viewpoint is obscene. For those that can't hike up, there can be a shuttle. Wider roads = more cars = just as much congestion, more pollution, and worse conditions for cyclists, not better. A few painted stripes and bike stencils on the road won't change that.

So please ride East Road. Ride it often. Fortunately, I wasn't the only one doing so: I saw a number of other cyclists doing the same. We as cyclists don't need to be coddled any more than motor vehicle operators need to be coddled. If anything, we need to be coddled less: cycling, unlike driving, is a right, not a privilege.

Saturday, April 24, 2010

predicting Old La Honda running time

Here's the model I described, fitting an equation to Minetti's data:

revised fit to combined run/walk dataRevised fit to combined run/walk data from Minetti

I should in theory be able to predict my time running up Old La Honda Road, assuming I am equally adapted to running as I was at cycling when I set my Old La Honda cycling PR last year.

From that ride I calculated this was around 299 watts as would be measured by an SRM, Quarq, or Metrigear Vector, or 5.31 W/kg. So what if
I could sustain that for the somewhat longer time I'd need to run up the hill?

First, I calculate the energy needed to run up Old La Honda. The simple way would be to take the average grade and use that, but instead I divided the climb into segments using iBike data and calculated the work required for each segment.

This work neglects wind resistance (it's from treadmill data) so I need to add that. Given a height of 1.69 meters, a width of 40 cm, and a CD = 0.8 with ρ = 1.15 kg/m³, I can calculate wind resistance as a function of speed: Pw = ½ ρ CD A v³, where A is my cross-sectional area (0.68 m²) and v is my speed. Since wind resistance power affects speed, and speed determines wind resistance power, the solution needs to be done self-consistently. I assume I weigh what I did when I did the Old La La Honda PR, and I added 3 lb for shoes and clothes. I used a Perl script for the calculation.

The result: it predicts I could run up Old La Honda in 23:38, a pace of 4:23 per km. Now I can state with 100% certainty I couldn't run that now. I view a prediction like that what a proto me, not having taken up cycling but running instead, could have done. Or maybe not even: maybe the same individual, similarly trained, can't produce as much power running as cycling.

In contrast, Gary Gellin ran a 24:29 during the first year of the Low-Key Hillclimbs. Gary's always been a bit faster than me riding uphill, and is a much better runner. So this reinforces the optimism of the calculation.

One possibility is running on a real road surface is less efficient than running on the treadmill used by Minetti. Another possibility is riding about allows for more work production (either greater metabolic exertion or a higher efficiency of conversion of metabolic exertion to mechanical work). At least the first issue should be evident in examining additional studies, as the issue of the energy cost of running has been investigated before: Minetti's work was unique in the range of road grades used.

Friday, April 23, 2010

New Balance shoe preview

I attended a nifty New Balance event at Zombie Runner on California Ave in Palo Alto yesterday. Anton Krupicka and Erik Skaggs joined a 4-mile group run from the store onto Stanford campus, then New Balance reps showed off shoe prototypes and promoted the MT100, New Balance's recent replacement to the "cult favorite" 790, which is what I wear.

Anton on left, Erik second from right, from Mark Tanaka's report on the 2008 Quad Dipsea

Anton and Erik are the two sponsored runners who were used as the model for the design of the 100MT. A nice review of that shoe is here. New Balance engineers gave Anton and Erik shoes, had them run a bunch, then put more padding where they wore the shoes down, removed padding where they didn't. So if you run like Anton and Erik then these shoes are great for you.

But then then had more shoes directed towards the minimalist aesthetic. Vibram 5-fingers shoes (more like foot gloves) are selling like crazy. New Balance's prototypes are aimed at capturing some of the same market: minimalist uppers with only 4 mm of heal-to-forefoot difference (as opposed to 10 mm in the old 790 and the new 100MT). One model, like the Vibram, was laceless, but a model preferred by Anton and Erik had traditional laces. I'll definitely try some of these when they're available next year: the Vibram toe pockets seem a bit gimmicky to me, especially since I prefer a bit of toe protection as I tend to slam my foot into obstacles on the trail when I'm looking too far ahead.

Then there was also the MT101, an alternate to the MT100. It's a bit heavier, with more support on the side of the foot, and without the "rock plate" on the fore-sole. Anton said he doesn't like limiting himself to a single pair of shoes, but chooses different shoes for different conditions. So New Balance doesn't consider the choice between the 100MT and 101MT to be exclusive.

Good stuff. But the real highlight was the group run. Even only four days out from Skyline to the Sea and having ridden the Noon Ride earlier in the day, I felt fairly good. Anton and Erik ran shirtless, commenting on the heat. Funny: Anton's from Colorado and Erik lives in Oregon. To me it felt cool.

My dinner for the day was three Bonk Breaker Bars, which were being sampled. Really good stuff. Not too sweet at all, with a natural ingredients taste. I'll definitely be buying more of these for long bike rides.

Thursday, April 22, 2010

Skyline to the Sea trail marathon

Running this year has been a bit of an adventure for me. I'd planned last winter to the Austin Marathon in February, but I had to put that aside when several bouts of sickness following a RedSpokes bike tour to Southeast Asia set me too far back in my preparation. Instead of the marathon, I did the Woodside Half in Huddart Park, an Envirosports trail race, my second trail race that season. I loved it. So this fall/winter, when I started running again, my focus returned to trail runs.

A friend told me about how she'd planned on Skyline to the Sea, but couldn't go. Maybe I'd like to try it, she suggested. Skyline to the Sea is a rare thing in trail runs: a point-to-point race. Point A to point B. It doesn't get any more fundamental than that. I had to try it.

There's two routes: the more popular "50 km" course includes an added loop up and down a hillside. The simpler route is claimed to be a marathon: slightly longer than the 26 miles + 385 yards which defines that distance. For some reason I felt a hilly trail marathon would be easier than a road marathon. A trail, I reasoned, breaks things up, uses different muscles, and reduces the pressure of maintaining a target pace.

Preparation wasn't great. My issue was I'd raced two long trail races in the past two months: the Coastal Trail Runs Rodeo Beach 18-miler (with distance run at the end to make up for a short-cut I inadvertantly took) and the Pacific Coast Trail Runs 30 km race at Pirates Cove. Trail runners tend to do races as training by backing off on the pace, focusing on building endurance without digging themselves into a hole. I wasn't worried about holes: I ran each of these races to get the best results I could. Take what you can get was my philosophy. And I was pleased with how I did at the first race (until the wrong turn) and at Pirates Cove: decent results; an age-group ribbon.

A result was I spent a lot of time recovering from these runs. Two weeks from Rodeo Beach. Two weeks (although with a lot of cycling in there) at Pirates, as well. This was time I could have spent working more on endurance.

Leading to Skyline to the Sea I had a decent block of work. Runs of 11.2, 12.3, 9.5, and 10.0 on alternate days over a 7-day period. Then on a trip to visit family in New Jersey 2.5, 5.6 (with a 7 mile hike after), and 6.8 miles on consecutive days. The promising thing was I felt better as this block progressed. These runs gave me some confidence. The issue was none of my runs, on their own, had really prepared me for the considerable jump from 30 km to 43, the marathon distnce.

The long day of trains and plane from New Jersey to San Francisco left me depleted, however. Allergies left me tired in the five days between my trip and the race. I didn't do much during this time: a few short bike rides and, the day before the race, an easy 1.5 mile run. I'd done what I could and I had to rely on the stimulus of competition to get me through the big day.


Pacing was going to be really important. The run starts, we were told, with some single-track on which it would be difficult to pass. I wanted to be positioned in the the third row: the top 20 but not the top 10 runners. I didn't want to go off and try to match the pace of Leor Pantalat or Scott Dunlap. But on the other hand I didn't want to get stuck in slow-climbing traffic on the "rollers" which characterized the opening kilometers of the race.

But at least from my position, concerns about the "pack" were relatively unjustified. Things strung out quickly. I felt as if I were running comfortably, somewhat briskly but without strain. On the climbs I shuffle-steeped, avoiding the inefficiencies of a full run. On the descents, I did the best I could with my short stride. Stay loose, stay comfortable. I lost time on the descents relative to those around me, but seemed to gain it back on the climbs.

After cross-crossing Highway 9 a few times, I arrived at the 10.5 km aid station at Waterman Gap in 55 minutes. This was ahead of my 10 minute per mile target, but then there was serious climbing to come, so I expected to be ahead of pace by this point. I decided to save the Coke option for later to avoid burning out on it, drinking some Clif carbohydrate solution and water, refilling my now empty bottle which I carried on a belt (I also had caffeinated Powerbar GelBlasts, which I'd occasionally eat), ate a few Clif Bloks, and set off for the climbing I knew followed ... climbing I looked forward to.

The climb wasn't bad, but despite this, as I reached the top I was fatiguing. Not good. Still close to 30 km remaining, a distance I had only run on trails twice this year, and I was already tired. Of course I'd been tired all week, so lasting 13 km before fatiguing was an improvement. But 29 km is a long way to go, especially over challenging terrain. I decided I needed to really throttle back from what I'd already thought was a sustainable pace. Take it one hour at a time, thinking about turning the throttle back up after three hours.

And this helped. To my relief, the second hour passed fairly well. The next aid station, near China Grade Road, I drank some Coke and water, took a few more Clif Bloks, and again was back on the trail fairly quickly. And from here it was a relatively short, albeit hilly stretch to the third aid station, near park headquarters.

Along the way I managed to take one slightly wrong turn: I was running along China Grade road, which seemed wrong. I asked some passing hikers, and they directed me back onto the trail which paralleled the road. I was quickly back on track. This probably saved me a few seconds (fewer than 10) versus the trail: deja-vu from my accidental short-cut at Rodeo Beach. But I wasn't about to backtrack at this point.

Onward. I eventually reached the next stop at the Park Headquarters. I was starting to feel depleted again. My Power Shots were mostly gone, and I'd started dipping into my supply of Enduralytes. I was drinking well, I felt, but now more than two hours in, I felt like I was fading a bit again. So at this station I stopped for a bit, ate some orange slices in addition to drinking Coke and water, then set out again.

Soon after I took another wrong turn: finding myself this time at the parking lot at the Park Headquarters. I backtracked and got back on the route, which was flooded and thus had skipped my attention, deciding this was karmic accounting for my earlier mistake: payback of those few seconds gained with interest.

What followed was the most challenging part of the course. The path climbed over rocks, went up and down steep slopes, and generally made itself difficult. Normally all good stuff, but I found myself no longer power walking, but more like trudging up climbs, and carefully stepping over obsticles to reduce fatigue-causing impacts.

Hour three arrived as I was passing a hiking group from Team in Training. I asked what they were training for and two responded: one for the Grand Canyon and the other for a different hike. They were inspiring, but not enough to convince me to pick up my pace as I'd planned to do at this time. I decided to wait for the fire road and reevalute.

The fire road, the fire road! Just make it to the fire road which marked the final 10 km, I told myself. If I could make it to the fire road, everything would be good.

I managed to pass the waterfall which is the highlight of the Skyline to the Sea trail without noticing it. I was too focused on continuing my progress to the fire road and, from there, the finish. One step at a time. I was running okay on the flatter parts, walking on the climbs and descents. It seemed to go on and on and on.

A runner passed: one of several in this segment. I mentioned to him that surely the fire road was soon. He cautiously agreed.

And we were right. The trail opened up and volunteers pointed the direction home. "5.8 miles", one said, to the finish. I asked how far it was to the aid station. I'd forgotten from my study of the course match back up on Skyline. He didn't know.

I started to run, but my motivation to do so, my willingness to suppress the pain, wavered, hovering at the critical threshold, and then snapped. I simply had to walk. Just for a bit, I told myself. I'd walk until I'd recovered a bit, then slowly run it in.

But my exeriments in running all failed. Each time, I simply wasn't able to refind that rhythm to get me through the pain. It became clear this was turning into a little death march. Just keep moving, I told myself. Keep walking.

I was passed repeatedly here. Many on the 50 km route, some marathoners. Caitlin Smith, the 50 km women's winner and third overall, had passed me well before the fire road, happily telling me I looked good. But by this point, on the fire road, I dreaded these wishes. I just wanted to fade into the background. I hoped those passing would assume I was a day hiker. Too many did not.

I asked hikers and cyclists (which are allowed on the fire road section) how far it was to the aid station. Nobody knew. Nobody seemed to have seen it.

Finally a numberless, oncoming runner told me the aid station was just ahead. This perked me up: finally, relief. I was there soon enough. A volunteer cheerfully told me it was only 1.5 miles to the finish, but there was no way I wasn't going to stop and recover a bit. 1.5 miles can be a long way. So I ate pretzels, drank some carbohydrate solution, and even ate half a frosted Pop-Tart. I'd hoped the Pop-Tart would revive me, but it only made me wish I'd eaten Clif Bloks instead: the Pop-Tart was too sweet, too pasty. I wondered if I'd dropped it if it would decompose.

There was nowhere to sit at the aid station, which was a good thing: it got me back on the trail sooner. I really wanted to run it home from here, but again I was unable. So I walked: head down and fairly miserable. When I reached the finish (18th / 54), I felt no sense of accomplishment, no victory, just relief.

I found Juliana, who had given me a ride to the start and was here to support Bjorn, her friend. She was really positive, helping to raise my spirits a bit. The important thing, she said, was that I'd finished, and this was a good benchmark for the next time I did it. Next time.... usually at the end of a hard event I don't like thinking about a next time. But there was no doubt in my mind there would be a next time. I had to return: return and get it right.

post-race, chewing PowerBloxPost-race, chewing PowerBlox

As I lay on the ground, munching PowerBlox (whey protein + sugar), I overheard a woman say this was the hardest marathon course she'd ever run. That helped a bit.

So what did I need to get right? First, start a bit more conservatively. But really my problem was just lack of fitness: lack of the specific adaptations which allow cardiovascular endurance to be applied to running. I got 80% there: I just needed that extra 25% which would have had me succumb past the finish line rather than 5 miles before it. A month of solid training and I certainly would have had it. Not that I could have run more given the date of this run: you've got to follow what your body lets you do. But next time I'll have more preparation.

Three days later, I'm surprised how well I've recovered. No real workouts yet, although I can run up stairs and rode my bike to and from the train commuting to work. I'm ready to start doing some training rides again. Easy spins, nothing hard, but it's good to know I didn't dig myself into too deep a hole at Skyline to the Sea.

Thursday, April 8, 2010

running power, part 5 (Minetti data revisited, and cycling vs. running)

Last time I described Minetti's data for running and walking on road grades extenting from ‒45% (descending) to +45% (climbing). After converting grade (the tangent of the angle) to the sine of the angle (distance climbed per unit distance traveled), I applied a heuristic model based on the assumption that in the asymptotic limit both climbing and descending involved an energy cost proportional to the altitude gained or lost.

It's time to reconsider that model...

First, Minetti shows for every grade tested, walking is more efficient than running. So then why not always walk? Well, obviously human kinetics limit the speed at which walking retains its efficiency, so beyond a certain speed, running is the preferred choice.

Really, I don't care about the cost of running a 45% grade. In a race, I'd never run a 45% grade: at whatever speed I can run that, I can certainly walk with less energy cost. So to estimate at what grade walking becomes preferred, I'll make some crude estimates. First, I assume I can gain altitude, best case, at around 1000 meters / hour. Then I'll assume I can walk up to 8 km / hr (around 5 mph). The grade, then, at which I can walk at my maximum speed is then 1 / 8 = 12.5%. I'll therefore assume I run if the climb is less steep than 12%, walk if it's steeper than 12%. I'll combine Minetti's data (walking > 12%, running < 12%) into a single data set, and refit.

So back to the fitting equation. I had a transcendental equation last time I thought was really slick and which I really wanted to work. Problem is: it didn't work. The fitting parameters in the model weren't matching what was expected from the underlying physics (most notably, the intercept was way too negative). So I decided that maybe Minetti was onto something with his polynomial. The differences in my approach:
  1. I fitted the polynomial to my g', the sine of the angle, rather than to grade, since g' is a more natural unit relating climbing rate to speed on the ground, and
  2. I made sure the highest order of the polynomial was even, as opposed to odd, to prevent it from exploding in the negative direction for sufficiently steep descents.

The result is shown in this plot:

revised fit to combined run/walk dataRevised fit to combined run/walk data from Minetti

Also shown in the plot is an "ideal" curve in which I assume all useful work goes into raising altitude. I set the metabolic efficiency to a relatively high 25.6% to prevent this ideal curve from moving above the fitted curve: walking can't be more efficient than "ideal". You can see the data, here in the walking regime, approaches the ideal curve but then rises above it. At some point it's faster to walk further up a more gradual grade than to take a short-cut up a steeper grade. This is consistent with everyone's experience, I'm sure.

Following a suggestion by Gary Gellin, I then I took the ideal curve and assuming the same metabolic efficiency applies to cycling, I added in additional power for drivetrain losses, added weight of the bike (assuming a bike weighing only 10% of total body + clothing weight), and a coefficient of rolling resistance of 0.5% (high-pressure road tires on a slightly rough road). Interestingly, this curve crosses the running curve at around 15.7%. This suggests if an athlete is similarly adapted to running and cycling, then it's faster to run (rather walk) a hill with a grade over 15.7% than it is to ride it, even assuming the bike has sufficiently low gears to allow for an efficient cadence.

Cadence is an interesting issue. For example, this study shows that cycling can have a very high metabolic efficiency (30%) at 50 rpm, but it drops at higher cadences, although riders had increased endurance at higher cadence. So focusing too strongly on efficiency alone is a mistake.

In this run-bike comparison, I neglect wind resistance, which I assume will be similar for the runner and the cyclist. This assumption is likely favorable to the runner, who is more upright, but also lacks a bike, which is typically around 1/3 of total wind resistance.

Next time, I'll compare running on the flats to cycling up Old La Honda Road, following Gary's observation that world class 5 km times may be a good predictor of what a world-class cyclist would do up Old La Honda (5.4 km @ 7.3%). Then I'll look at the benefit of drafting in running.

Tuesday, April 6, 2010

running power, part 4 (Minetti data)

Philip Skiba developed metrics ("GOVSS"), comparable to the "training stress balance" (or his version, "BikeScore", which has been implemented in Golden Cheetah) used in cycling, for running to allow multi-sport athletes to extend their training stress calculations across the two sports. Since runners generally don't have power meters, it becomes necessary to estimate training stress using speed and altitude profile. For that, Skiba taps into the work by Minetti and coworkers published in 2002 in the Journal of Applied Physiology.

The authors first measured base metabolic rates of subjects at rest as a baseline. Then they walked or ran on a treadmill at various degrees of inclination, extending over a range of grades from ‒45% to +45%. Metabolic rate was measured in each case and found to be proportional to the speed, but with nontrivial function of slope.

In cycling, it's relatively simple: at a given speed, there is a power component proportional to linearized slope (grade / √1 + grade²). But running is different. In cycling, you can coast downhill, whereas when running down hill, your muscles are doing work no matter what the grade. I was going to propose an analytic form for this relationship purely on conjecture, but thanks to Andrew Coggan's tip, I found Minetti's experimental data.

Here's a plot of Minetti's result, the energy per unit distance consumed on the various grades, along with equations I fit to his data. Minetti provided his own fit, a sixth-order polynomial, but I didn't like that much as polynomials tend to have catastrophic extrapolation characteristics. Plus, my formula has only four fitting parameters in contrast to Minetti's six, and fewer fitting parameters is generally better. Skiba uses Minetti's polynomial; perhaps I can convince him to switch over to my formula.

Minetti data fittedMinetti treadmill data, with my analytic fit

The result is that for downslopes, the steeper the slope beyond a certain level, it becomes more, not less, difficult to run a certain distance. This certainly speaks to the relative difficulty of hilly trail runs: they add difficulty both going up and coming down.

Another interesting, but completely predictable result of this plot is that walking is always easier than running. That means at a given speed if you can walk, do it. This was advice given to me by Don at Zombie Runner in Palo Alto, and I've taken it to heart. On steep climbs I try to walk and find, doing so, I'm able to stay in the top 10% on climbs against people who are doing extra work bouncing around but not going any faster.

Now there's a tricky issue with this, which is that it's for metabolic energy, not work done. Cycling power meters measure work done. If the human body can produce work at a given fixed efficiency, then the two are interchangeable, with an efficiency typically reported to be around 22%. However, the human machine isn't so simple. It would be a mistake to use the same efficiency to convert these running and walking metabolic costs to work.

Fortunately, the approximation of fixed efficiency may be relatively better for cycling, which is an optimized interface between the body and the road. For example when riding a geared bike on different slopes, we can shift gears to keep the force on the pedals and the rate of pedal rotation similar to what we encounter on a flat road. So for the purpose of comparing cycling to running, these data might be suitable, as long as we convert the cycling work done (as reported by power meters) to metabolic energy.

Sunday, April 4, 2010

running power, part 3 (climbing)

Last time, I described how it's not so simple to come up with a model for the power required in running without a detailed analysis of human body kinetics. Bicycles are so wonderful because they act as an interface between human kinetics and propulsion. Assuming the bike fits the rider, the details of human kinetics cease to be important, and the simple power-speed relationship of the machine versus the road are simply analyzed.

Ed Coyle's model as described by Ed Burke is simple: 1 kcal per kg per km traveled (assuming standard Earth gravity). But obviously that's simplistic. Running up the World Trade Center stairs, for example, obviously takes more energy than running a similar distance on level ground.

The record in the Empire State Building Run-Up is held by Australian Paul Crake, in 9:33 in the 2003 race. That corresponds to a VAM of 2010 meters/hr for 9 min 33 sec. That's 0.558 meters/second.

Then this VAM corresponds to running 1.675 m/sec = 1.675 W/kg. Lifting COM to that altitude is 5.47 W/kg. If clothing is 2% of total weight, then that's 5.58 W/kg.

Again, Coyle's model for running energy used is 1 kcal/kg/km. Assuming a metabolic efficency of 23%, that's 1 kJ/kg/km. However, this number includes some assumption for wind resistance, which I prefer to model explicitly. Assuming A = 0.68 meters² (40 cm by 1.7 meters) and CD = 0.8, with air density running at 250 meters/minute, that yields 0.04 kJ/km/kg due to wind resistance. So I could reduce Coyle's number to 0.96 kJ/km/kg, a correction which is smaller than the precision of the obviously crude estimate.

So now I simply need to estimate the distance traveled by the runners. The stairs appear quite steep:

Empire State Building Run-Up

Assume for example the runners go 2 meters for every meter uphill, including landings. Then this VAM corresponds to running 1.12 m/sec = 1.07 W/kg. Adding in those 5.58 W/kg brings the total to 6.65 W/kg.

Wind resistance is another factor, of course. Again assuming CD = 0.8, A = 0.68 meters², M = 70 kg, and air density ρ = 1.2 kg/m³, that's 0.013 W/kg, bringing the total to 6.66 W/kg.

Curiously, that's close to the 6.7 W/kg claimed by Daniel Coyle (related to Ed?) as the lactate threshold needed to win the Tour de France. This race is a lot shorter than the one hour or so one can sustain efforts at near lactate threshold. On the other hand, the Empire State Building Run-Up doesn't have the prestige of the Tour de France, either, and the participants don't have the luxury of the preparation or training available to Tour contenders.

The issue with simply adding potential energy to Ed Coyle's 1 kJ/kg/km, however, is that it predicts, as is the case with a bike coasting downhill, that downhill running might be accomplished with zero power. Of course this is far from the truth. So something more is needed.

I'll conjecture on that next time.

I'm doubting Coyle's model applies to climbing. I suspect energy efficiency is better uphill: less bouncing. Hard to believe running would allow for more power output than cycling.

Saturday, April 3, 2010

running power: part 2

Last time I came up with an analytic model for the power "wasted" in running: the power required to bob the center of mass up and down, and to life the feet and calves during each foot strike.

Unfortunately, like so many models, this one is almost useless. There's two big problems:
  1. There is no predictive model for α, the fraction of time both feet are off the ground, and
  2. there is no predictive model for hfoot, the height to raise the feet, and
  3. the optimal cadence calculation is simplistic.

Now it's true my calculation of optimal cadence matched precisely my preferred cadence, that's only because I set α to get this result. α = 70% is a fairly high fraction of the time off the ground, and the result of over 200 "wasted" watts (watts in excess of those you'd get on a bicycle, where these power terms are not present) is obviously a high number.

The model predicts as α drops, optimal cadence drops in proportion. But of course this isn't true in practice. α = 0 corresponds to walking, and you don't walk with zero cadence. The problem is I set a fixed value of hfoot, the amount to raise the feet. Simple physics suggests hfoot should be as small as possible: drag the feet skimming the ground. But the Kenyans don't run that way in the photo, so obviously there are inefficiencies to running like that which I am not capturing.

The late Ed Burke says that according to Edward Coyle, runners use around 1 J/kg/meter, assuming standard gravity of 9.8 m/sec², and assuming a metabolic efficiency of around 1 J / cal. However, this neglects wind resistance and the effect of gaining or losing altitude. So these need to be added.

Wind resistance is easy: it's the same for a runner as it is for a cyclist.

Altitude is another matter: not so obvious. For a cyclist, there is a terminal velicity on a downhill of a given slope at which the forces are in balance, and the rider will coast at a steady speed. For a runner, this obviously is not the case. Running downhill may be easier than running on level ground, but it takes work, no matter how steep the slope. Similarly, running uphill is clearly more effort than running on the flat. But it may be difficult to calculate how much harder: suppose when running, the center of mass rises, then the center of mass falls. It's possible running uphill is more efficient, for example.

I can model this by changing the relationship between α, C, and t. The body spends less time dropping than it does rising, because it's rising more than it is dropping when running uphill. But modeling the center of mass relative to the ground instead of relative to a fixed altitude, we can include the power going into the "climb" separately, and the time rising versus falling relative to the ground is again the same. The equation becomes:

h = ½ g t² ‒ v t1 grade,

where v is the runner speed, and grade is the linear approximation to the road grade. I can then proceeed as I did last time. But what is already seen in this equation is that when running uphill (positive grade), h will be less, and efficiency improves, while for running downhill, will be more, and efficiency decreases, all for the same time spent ballistic.

But I'll also need a model for hfoot. That may change with road grade. And I don't have that model, so it's hard to proceed further. More information is needed.

The Coyle model as represented by Burke is obviously oversimplistic. But running's a lot more complex than cycling, and taking it further is a nontrivial task, to say the least.

Friday, April 2, 2010

running power: part 1

Last time on this subject, I introduced the idea of modeling the power required for running. Time for some equations.

Suppose I "run in place", bouncing up and down. When I'm in the air, my trajectory is determined by gravity, out of my control. So if I want to move my center of mass upwards, the more I move it up and down, the less often I can move it. "Cadence", which is the rate at which each foot contacts the ground, is directly related to the motion of my center of mass. So if I run at a particular cadence, and am in the air a given fraction α of the time, then the amount of energy it takes to raise my center of mass each bounce depends on the cadence and on α.

Consider the bouncing ball example. Flying upward under the influence of gravity with acceleration g, simple integration yields that the relationship between its maximum haight h (the vertical displacement of the center of mass) and the time t is:

h = ½ g t²,

where t is the time taken for the ball to reach its maximum height. The amount of potential energy gained during this trajectory is then:

ΔU = ½ M g² t²,

where M is total mass.

As a simple model, I'll assume the body bounces like a ball twice per stride, once on the left, the other time on the right, and spends a fraction α of that time in the air. Then cadence and t are related:

C = α / (4 t ),


t = α / (4 C ).

Assuming the power to raise the center of mass through potential difference ΔU comes from the muscles, and this must then be done twice per foot-stride, one gets:

Pcom = C M g² t²
= C M g² α² / ( 4 C
= M g² α² / 16 C.

This implies the faster the stride, the less the power required in raising and lowering the center-of-mass.

But then consider the power requirements of raising the feet. Assume the foot is raised to a height hfoot twice per foot stroke. Then assume the effective mass of the foot is Mfoot. This mass includes mass from the rest of the leg, in proportion to the amount each segment of the leg is raised. I assume the leg is raised the same amount independent of the cadence (this superposed on the center of mass displacement). Then the power requirements from raising and lowering the foot are:

Pfeet = 2 C Mfoot g hfoot.

To minimize this, you want as low a cadence as possible. Thus total power associated with bouncing around and raising and lowering your feet becomes:

Ptot = Pfeet + Pcom.

To minimize this is a simple exercise in differential calculus. The result:

optimal C = sqrt[ ( M / Mfoot ) g / 2 hfoot ] α / 4

Kenyan RunnersKenyan runners really kick up their feet on the back-stride

I laid down with my knee on the ground and my foot on the scale and got 6.8 kg. Adding on 320 g for my shoe and I get Mfoot = 7.1 kg. This is actually just what I want, because it calculates the effective weight of my calf and foot if I pivot at the knee. That's not a bad approximation for what happens when I kick while running. The thigh contributes as well, but I'll neglect that. Then I use for my body + clothing M = 57.6 kg. I'll guesstimate hfoot = 0.7 meters. If α = 0.7 (spending 70% of the time flying through the air), then I get an optimal running cadence of 94 strides/minute, which is what I generally run.

At the optimal cadence, with the optimal tradeoff between the two power components, the two power components are equal:
Pcom =
Pfeet =
g sqrt[ 2 Mfoot M hfoot g ] α / 4,

Thus the sum is:
Ptot =
g sqrt[Mfoot M hfoot g / 2] α,

which, using my parameters, yields an optimal power running at that cadence of 94 Hz of 218 watts.

So this simple model is really interesting: it says if I run along, my center of mass going ballistic around 70% of the time, each foot pivoting around its knee to a height of around 50 cm, I end up doing around 218 watts of power, independent of how fast I go. This fails to account for the additional work pushing my feet along the ground, propelling my body forward. So what am I missing?

What I'm missing is I arbitrarily estimated α and hfoot. If I run slower at the same cadence, lifting each foot up the same amount, I'll spend less time ballistic, and α will drop. If I can keep at least one foot on the ground at all times, then in principle I can keep my center-of-mass at a constant height, and I reduce the amount of power required. Additionally, at slower speed I don't need as long a stride, and I can kick up less, reducing power further by reducing hfoot. So really this model applies to only a single speed.

More next time.

Thursday, April 1, 2010

Powertap rewired!

My off-season (well, not so off-season at this point) focus on trail running this year has really killed my legs. As a result, I've been generally suffering on Old La Honda during the Wednesday Noon Ride. Guys I am accustomed to staying with have been simply riding away from me, chatting in a relaxed voice, while I suffer, gasping, in their wake.

But now that's going to change.

It occurred to me during my recent Pirates Cove Trail Run that my Powertap, which as I've documented hasn't been reporting accurate power anyway, can be put to a far better use. The key is the piezoelectric effect. The Piezoelectric effect, as applied to measuring power, results in the following relationship:

voltage = (torque ‒ τ0) × K,

for some τ0 and K. However, trivial algebraic manipulation yields:

torque ‒ τ0 = voltage / K,

where torque ‒ τ0 is "useful torque" sufficient to overcome the built-in tension in the powertap hub.

What this says is if instead of forcing a certain torque and measuring the resulting voltage, if I force the voltage I will instead generate a torque. This idea is hardly new, as it's the basis for piezoelectric motors.

Despite my overeducation in electrical engineering, I'm not very good with electronic hackery. But this mod was just too simple. A simple swap of connectors, and I was, you might say, ready to roll:

powertap mod
Simple Powertap mod

I weighed the hub before and after on a vibration-controlled microbalance in the class-100 clean room at the Stanford integrated circuit processing facility (don't tell!), and with my somewhat sloppy soldering technique, this added 47.6 mgrams, which adds around 0.4 msec to my Old La Honda time. But the payback is worth it.

Next there's the battery issue. Sure, the Powertap comes with a battery, but an actuator obviously demands more capacity than a sensor. The "obvious" approach would be to wire a connection to a frame-mounted battery. But a wire from the rear hub to the frame isn't the best approach, and in any case USA Cycling rule 1M(c) forbid the use of any "stored energy" for propulsion.

Fortunately it pays to have friends. In this case a grad student I know at Stanford is doing his PhD in "energy harvesting". This basically involves using sensors to harvest vibrational energy and converting it to useful electronic power. I lent him my wheel for a week, and when it came back, he'd cleverly installed silicon-based microsensors inside my moderate deep Reynolds rims (in return, I act as a test subject for his thesis, and help him with data analysis). The spokes act as perfectly useful wires with a bit of epoxy in the spoke holes as insulation on the "power" spokes (radial non-drive side). The non-drive-side, still touching the rim and hub, act as ground. This worked so well only because the Reynolds rim has the nipples internal to the rim, rather than extending from the rim, making the installation relatively straightforward. The weight penalty was a bit more here: 17.5 grams on my Scout, sensors + epoxy. But oh-so-worth it.

Okay, so in theory if you ride on perfectly smooth roads you don't get anything out of this. But the roads are never perfectly smooth, and on my favorite climb, Old La Honda, even with the relatively recent road work, I'll be than adequately powered for the task. The real beauty of this is if I stop, the vibrations cease, and the power shuts down. To get it going again I simply need to start the bike moving. No energy "stored and released": direct generation from road vibrations to voltage to torque to climb-slaying power.

So there it was. Did it work? Of course: the physics is just too simple. And the result? Let's just say based on preliminary tests in the Marin Headlands yesterday I'll be posting a new PR report on Old La Honda soon. Watch out, Diablo Hill Climb Time Trial in June!