Froome’s data on Strava

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Chris Froome has been logging data on Strava since the beginning of the year. He had already completed over 4,500km, around Johannesburg, in the first four weeks of January. The weather has been hot and he has been based at an altitude of around 1350m. Some have speculated that he has been replicating the conditions of a grand tour, so that measurements can be made that may assist in his defence against the adverse analytical finding made at last year’s Vuelta.

Whatever the reasons, Froome chose to “Empty the tank” with epic ride on 28 January, completing 271km in just over six hours at an average of 44.8kph. The activity was flagged on Strava, presumably because he completed it suspiciously fast. For example, he rode the 20km Back Straight segment at 50.9kph, finishing in 24:24, nearly four minutes faster than holder of the the KOM: a certain Chris Froome. Since there was no significant wind blowing, one can only assume he was being motor-paced.

One interesting thing about rides displayed publicly on Strava is that anyone can download a GPX file of the route, which shows the latitude, longitude and altitude of the rider, typically at one second intervals. Although Froome is one of the professional riders who prefer to keep their power data private, this blog explores the possibility of estimating power from the  GPX file. The plan is similar to the way Strava estimates power.

  1. Calculate the rider’s speed from changes in position
  2. Calculate the gradient of the road from changes in altitude
  3. Estimate air density from historic weather reports
  4. Make assumptions about rider/bike mass, aerodynamic drag, rolling resistance
  5. Estimate power required to ride at estimated speed

Knowledge is power

FroomeyTT

An interesting case study is Froome’s TT Bike Squeeeeze from 6 January, which included a sustained 2 hour TT effort. Deriving speed and gradient from the GPX file is straightforward, though it is helpful to include smoothing (say, a five second average) to iron out noise in the recording. It is simple to check the average speed and charts against those displayed on Strava.

Several factors affect air density. Firstly, we can obtain the local weather conditions from sources, such as Weather Underground. Froome set off at 6:36am, when it was still relatively cool, but he Garmin shows that it warmed up from 18 degrees to 40 degrees during the ride. Taking the average of 29 for the whole ride simplifies matters. Air pressure remained constant at around 1018hPa, but this is always quoted for sea level, so the figure needs to be adjusted for altitude. Froome’s GPS recorded an altitude range from 1242m to 1581m. However we can see that his starting altitude was recorded as 1305m, when the actual altitude of this location was 1380m. We conclude that his average altitude for the ride, recorded at 1436m, needs to be corrected by 75m to 1511m and opt to use this as an elevation adjustment for the whole ride. This is important because the air is sufficiently less dense at this altitude to have a noticeable impact on aerodynamic drag.

An estimate of power requires some additional assumptions. Froome uses his road bike, TT bike and mountain bike for training, sometimes all in the same ride, and we suspect some rides are motor-paced. However, he indicates that the 6 January ride was on the TT bike. So a CdA of 0.22 for drag and a Crr of 0.005 for rolling resistance seem reasonable. Froome weighs about 70kg and fair assumptions were taken for the spec of his bike. Finally, the wind was very light, so it was ignored in the calculations.

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Under these assumptions, Froome’s estimated average power was 205W. The red shaded area marks a 2 hour effort completed at 43.7kph, with a higher average power of 271W. His maximal average power sustained over one hour was 321W or 4.58W/kg. There is nothing adverse about these figures; they seem to be eminently within the expected capabilities of the multiple grand tour winner.

Of course, quite a few assumptions went into these calculations, so it is worth identifying the most important ones. The variation of temperature had a small effect: the whole ride at 18 degrees would have required an average of 209W or, at 40 degrees, 201W. Taking account of altitude was important: the same ride at sea level would have required 230W, but the variations in altitude during the ride were not significant. At the speeds Froome was riding, aerodynamics were important: a CdA of 0.25 would have needed 221W, whereas a super-aero CdA of 0.20 rider could have done 195W. This sensitivity analysis suggests that the approach is robust.

Running the same analysis over the “Empty the tank” ride gives an average power requirement of 373W for six hours, which is obviously suspect. However, if he was benefiting from a 50% reduction in drag by following a motor vehicle, his estimated average power for the ride would have been 244W – still pretty high, but believable.

Posting rides on Strava provides an independently verifiable adjunct to a biological passport.

The fractal nature of GPS routes

The mathematician, Benoît Mandelbrot, once asked “How long is the coast of Britain?“. Paradoxically, the answer depends on the length of your measuring stick. Using a shorter ruler results in a longer total distance, because you take account of more minor details of the shape of the coastline. Extrapolating this idea, reducing the measurement scale down to take account of every grain of sand, the total length of the coast increases without limit.

This has an unexpected connection with the data recorded on a GPS unit. Cycle computers typically record position every second. When riding at 36km/h, a record is stored every 10 metres, but at a speed of 18k/h, a recording is made every 5 metres. So riding as a lower speed equates to measuring distances with a shorter ruler. When distance is calculated by triangulating between GPS locations, your riding speed affects the result, particularly when you are going around a sharp corner.

Consider two cyclists riding round a sharp 90-degree bend with a radius of 13m. The arc has a length of 20m, so the GPS has time to make four recordings for the a rider doing 18km/h, but only two recordings for the rider doing 36km/h. The diagram below shows that the faster rider will have a record of position at each red dot, while the slower rider also has a reading for each green dot.  Although the red and green distances match on the straight section, when it comes to the corner the total length of the red line segments is less than the total of the green segments. You can see this jagged effect if you zoom into a corner on the Strava map of your course. Both triangulated distances are shorter than the actual arc ridden.

Cornering.pngIt is relatively straightforward to show that the triangulation method will underestimate both distance and speed by a factor of 2r/s*sin(s/2r), where r is the radius of the corner in metres and s is speed in m/s. So the estimated length of the 20m arc for the fast rider is 19.4m ridden at a speed of 35.1km/h (2.5% underestimate), while the corresponding figures for the slower rider would be 19.8m at 17.9km/h (0.6% underestimate).

We might ask whether these underestimates are significant, given the error in locating real-time positions using GPS. Over the length of a ride, we should expect GPS errors to average out to approximately zero in all directions. However, triangulation underestimates distance on every corner, so these negative errors accumulate over the ride. Note that when the bike is stationary, any noise in the GPS position adds to the total distance calculated by triangulation. But guess what? This can only happen when you are not moving fast. The case remains that slower riders will show a longer total distance than faster riders.

The simple triangulation method described above does not take account of changes of elevation. This has a relatively small effect, except on the steepest gradients, thus a 10% climb increases in distance by only 0.5%.  In fact, the only reliable way to measure distance that accounts for corners and changes in altitude is to use a correctly-calibrated wheel-based device. Garmin’s GSC-10 speed and cadence monitor tracks the passage of magnets on the wheel and cranks, transmitting to the head unit via ANT+. This gives an accurate measure of ground speed, as long as the correct wheel size is used (and, of course, that changes with the type of tyre, air pressure, rider weight etc.).

According to Strava Support, Garmin uses a hierarchy for determining distance. If you have a PowerTap hub, its distance calculation takes precedence. Next, if you have a GSC-10, its figure is used. Otherwise the GPS positions are used for triangulation. This means that, if you don’t have a PowerTap or a GSC-10 speed/cadence meter, your distance (and speed) measurements will be subject to the distortions described above.

But does this really matter? Well it depends on how “wiggly” a route you are riding. This can be estimated using Richardson’s method. The idea is that you measure the route using different sized rulers and see how much the total distance changes. The rate of change determines the fractal dimension, which we can take as the “wiggliness” of the route.

One way of approximating this method from your GPS data is, firstly, to add up all the distances between consecutive GPS positions,  triangulating latitude and longitude. Then do the same using every other position. Then every fourth position, doubling the gap each time. If you happened to be riding at a constant 36km/h, this equates to measuring distance using a 10m ruler, then a 20m ruler, then a 40m ruler etc..

Using this approach, the fractal dimension of a simple loop around the Surrey countryside is about 1.01, which is not much higher than a straight line of dimension 1. So, with just a few corners, the GPS triangulation error will be low. The Sella Ronda has a fractal dimension of 1.11, reflecting the fact that alpine roads have to follow the naturally fractal-like mountain landscape. Totally contrived routes can be higher, such as this one, with a fractal dimension of 1.34, making GPS triangulation likely to be pretty inaccurate – if you zoom in, lots of corners are cut.

In conclusion, if you ride fast around a wiggly course, your Garmin will experience non-relativistic length contraction. Having GPS does not make your wheel-based speed/cadence monitor redundant.

If you are interested in the code used for this blog, you can find it here.