Thursday, November 29, 2012

Converting Run Times to Cycling Times for Low-Key Hillclimbs: Part 2

To convert times for running to an "effective cycling" time the approach I want to take is to convert the running time to a cycling time of similar metabolic cost. Minetti measured metabolic cost for running on a treadmill tilted at different angles. This completely neglects wind resistance, yielding the running analog of rolling resistance, overcoming gravity, and drivetrain efficiency:

For cycling I have the standard equations, which I won't fully enumerate here. But in still air there's a wind resistance component proportional to area, air density, and speed cubed, a gravity component and a rolling resistance component each proportional to weight (gravity and mass), speed, and the sum of the road grade and the coefficient of rolling. Additionally there is a drivetrain loss component which is typically if inaccurately modeled as proportional to total power (in reality the fractional loss decreases at higher power).

There's several levels of approximation I could use. First, there's the issue of wind resistance, Wind resistance depends on speed, and so if I were to carefully model wind resistance I would get a different conversion factor for each runner based on his speed. But this adds needless complexity given the uncertainty in the model parameters, so I'll assume typical values for running and cycling speeds to account for the wind resistance effect.

Then there is what to use for the road grade. One approach is to use aggregate statistics for the climb. But while that is certainly simple and would be a substantial improvement on status quo, I feel I should do better: there is a substantial difference between a climb and descent, with zero net altitude gained, versus a flat road with zero net and gross altitude gained. So I will analyze the road on a piece-by-piece basis. But road grade has something of a fractal character: the closer you look, the more it varies, so I will smooth the road by a 25 meter bi-exponential smoothing function first.

The approach is then the following:

1. Smooth the road grade with a bi-exponential smoothing function of length 25 meters.
2. For each segment of road, calculate a metabolic cost per unit weight per distance for running.
3. Estimate the time taken to run that segment as proportional to the metabilic cost multiplied by the distance.
4. Calculate the time taken to ride the same segment, in steady-state, under the assumption of constant propulsive power. For this I need an assumption of rider propulsive power. The time should be modified on steep descents since riders tend to brake or coast for safety reasons.
5. Sum up the course times for running and cycling.
6. Adjust times for speed lost from event duration for runner and cyclist (I'll discuss this later).
7. Calculate a ratio of running time to cycling time.
8. Apply a scale factor to establish a desired balance of running versus cycling scores.

The issue of metabolic efficiency is important. When I used a similar approach to estimate my running time up Old La Honda Road, I got a substantially optimistic result, faster than even Gary "the" Gellin was able to pull off during a Low-Key Hillclimb. The metabolic efficiency question is an important one.

But whatever I pick for metabolic efficiency, there is the additional question of score balance. I want Low-Keys to be a cycling event, not a running one, so cyclists should generally score higher then runners.

For the metabolic efficiency of running versus cycling, I'll bypass the issue to some extent. For running, I can assume metabolic cost per unit time is constant, and thus speed is inversely proportional to metabolic cost per unit distance. There is a calibration factor to be determined, but to get this I just need to assert a given speed on flat roads. For example, 4.2 meters / second corresponds to a 39.7 min per 10 km, or 1:23:43 per half-marathon. Then for each road-grade I self-consistently solve metabolic cost of wind resistance and speed. For the incremental metabolic cost of wind resistance I'll arbitrarily assume a 25% marginal metabolic efficiency. Since wind resistance is approximately anyway, higher precision is pointless. So the sequence is solve total metabolic cost under an assumed speed, revise speed based on total metabolic cost, repeat.

For cycling, I simply set a target sustained power of 5 watts / kg. But since riders are typically presented by technical challenges on steep downhills which require some braking, this approach overestimated downhill speed. I set an arbitrary speed limit of vmax = 20 m/sec, which is 72 km/hr or 44.7 miles/hr. After I calculated a speed under full power, I did a transformation to get a final speed:

v → (v−4 + vmax−4)−1/4 .
This adjustment results in a certain amount of technically-related speed loss at all speeds. For example at an ideal speed of 50 kph it reduces actial speed to 47.5 kph.

These calculations give me a best-effort prediction of the relative speeds of runners versus cyclists for each road grade, assuming running at that pace and cycling with that power are comparable achievements. However, to use this conversion factor directly would balance running and cycling scores. I don't want to do that, since I want this to be a series where the focus is on cycling, not running. So the obvious remedy is to scale effective running times by an additional factor.