Tuesday, July 13, 2010

Rating Climbs: some existing formulas

Fillmore Street in San Francisco: short and very, very steep.   Webcor Cycling

The last time I wrote on this topic I described some of the issues with rating climbs. Here I'll describe some of the existing formulas.

First a quick disclaimer. I tend to interchangably use grade and climbing/distance. By grade here I generally mean climbing/distance, even though grade is formally rise over run (not over the hypotenuse).

net distance
Let's start simple. One formula is distance. This seems absurd, but it's one you even hear today from commentators on bike racing. "Road xxx climbs for yyy km." Obviously distance provided difficulty, but we're interested here in the difficulty of climbing, and without considering road grade, distance is a very weak metric.

net climbing
Next is total climbing. As I noted climbing can be measured as net or gross climbing, but for simplicity I'll stick with net climbing. Then the total climbing is the initial altitude subtracted from the final altitude.

This actually is a fairly good metric if you assume the rider has an infinite range of gears, and rides a bike with a sufficiently long wheelbase that lifting the front wheel is never a concern, and the stability isn't affected. Whatever the grade, the rider can move along with a comfortable force on the pedals. From a simple energy perspective, the best way to gain a certain amount of altitude is in the shortest distance possible. Looking just at the climbing component, the energy requires depends only on the weight of the rider and all of his equipment multiplied by this altitude change. Other energy components, like rolling resistance and wind resistance, are less with a shorter path than a longer one.

The issue here is that most riders on racing-style bikes don't have an infinite range of gears, and the wheelbase is short enough that the wheelbase is short enough that less of the weight is supported by the front wheel, making it easier to lift that wheel off the road. Additionally, inertia is less, so the bike doesn't coast as well, making it more important to apply a continuous propulsive force on the pedals without dead spots. And when the bike it tilted upward, the intrinsic stability of the trail is reduced. To understand this latter effect, consider that when you tilt the bike to the right the front wheel wants to rotate into the lean, which will cause the moving bike to be restored towards being upright. But if the bike is vertical (wheels stacked one over the other), this effect is gone; gravity no longer causes the wheel to respond to changes in the bike's "roll". An additional effect is for the trail to provide stability, the bike needs to be moving forward, and thus bikes become less stable when moving very slowly. So considering all of these effects, if the hill gets steep enough, it can become very unpleasant to ride the bike.

climbing² / distance
So most formulas consider road grade in addition to altitude gained. For example, the formula used by John Summerson in his ratings (as opposed to what is described in his text) is to multiply the altitude gained by the average road grade. Assuming a linear approximation to road grade, this implies a rating proportional to altitude gained squared divided by distance traveled. John didn't invent this formula; it had been used for years to rate climbs in Europe.  John did add additional factors for finishing altitude, grade variability, and for whether the road contains unpaved portions, but I'll let you buy his excellent books for details on those.

The issue with this core formula is that it suggests, for example, that a 6% grade is twice the "difficulty" of a 3% grade for the same net altitude gained, which most fit riders would agree is not the case. this comes back to the use of net altitude to rate climbs: as long as the bike has adequate stability, gears are low enough to provide a suitably low pedal force, and the front wheel doesn't lift off the pavement during pedaling, climbing a certain number of vertical feet isn't particularly harder at a steeper grade than a lesser one. It's only when the grade reaches a certain threshold that it causes a problem. And beyond that threshold, it can become a really big problem.

climbing3/2 / distance1/2
John tried to strike a compromise in the formula described in his text (not the one used in generating ratings) by multiplying the square root of the grade by the altitude gained. This mitigates the issue of a 6% grade being rated twice as hard as a 3% grade: instead it becomes only 41% harder. But on the other hand, I think most people would agree that a 30% grade is at least twice as hard, for the same net climbing, as a 15% grade. So the square root doesn't quite do justice to truly steep roads.  It fails to recognize grade becomes an issue only when it gets steep.

So to address this more complex formulas have been proposed. John's formulas have the advantage that they have no parameters: altitude and grade are indesputable characteristics of the hill.  The rating doesn't need to "know" anything about rider fitness or gearing or bike geometry.  But sometimes a bit of complexity is merited in the name of accuracy.   Sometimes it's worth making "reasonable" assumptions about these sources of variability.

One such attempt is from Climb By Bike. Their metric is:

(H / D) × 400 m + H² / D + D / 1000 + (T ‒ 1000 m) / 100,

where H is the net climbing, D is the distance, and T is the peak altitude.

This formula has serious issues. First, consider the first term: (H/D) × 400 m. This is independent of how long the climb is. So a climb which gains 2.5 mm in 1 cm of distance scores 100 meters here. Another climb, which gains 100 meters over 2 km, would score only 27 meters total. So the formula suggests that climb of 2.5 mm over 1 cm is close to 4 times as hard as climbing 100 meters over 2 km. That's just plain silly.

Next, the final term. This gives a climb a fixed score simply for the altitude. Same issue: an absurdly easy climb starting from a few thousand meters of elevation could be rated higher than a quite challenging but short climb at sea level.

Both of these issues are due to a fundamental problem: these difficulty components (grade and altitude) should be multiplied with, not added to, terms proportional to altitude gained or distance traveled. A nice rule of thumb is if the climb is twice as long, gaining twice the altitude, the rating should double. These two "bonus" terms don't change with the length of the climb.

A final issue with the formula is it is giving a bonus for distance, which might allow that a climb of lesser grade is rated higher than one of steeper grade for the same net climbing. As I discussed in the previous post, such a road may be harder to ride, but that doesn't mean "the climb" is tougher. So I prefer ratings where a more gradual climb is never ranked harder than a steeper climb for the same net climbing.

The core of this formula is the one used by Summerson: H²/D. The attempt to improve that formula's deficiencies is laudible, but failed.

I'll describe my formulas in the next post on this subject.

1 comment:

Ron said...

Great to see you tackle climbing ratings. Interesting challenge.

FYI That post I had written on wind effects has received updates. http://cozybeehive.blogspot.com/2010/07/wind-altitude-their-effect-on-uphill.html It is a complex topic indeed, indeed non-linear.