Make Like a Tree and Get Out of Here

Swarthmore's Scott Arboretum traditionally gives incoming students a plant to care for in their dorms. Shockingly, the Hawaiian Schefflera I received in 2007 is still going 17 years later! Over the years as I've moved from place to place and kept the plant in different environments, I've been impressed by its ability to track the sunlight, frequently growing lopsided as it reaches toward the nearest window until I think to turn it. This has resulted in some twisting, gnarled branches:

If you've been reading this blog, you can probably guess where this is going – I was curious if I could make a simulation of my plant's heliotropic tendencies. I decided to model the plant as a collection of connected branch segments, each a fixed length and pointing at an angle relative to the vertical. At each step, we iterate over all the segments and pick an action:

• If the segment has no children, i.e. it's at the tip of a branch, we add a new segment on the end with a probability p_grow/size, where size is the number of existing segments.
• If the segment does have children, we add a new one with probability p_sprout/size.
• If neither of those occur, we adjust the angle of the branch to point closer to the sun's current position. The adjustment is proportional to how far off the angle is, how many branches are on the end of this one, and a constant stiffness for the plant.

I tried a bunch of values for the different parameters until I landed on a range that gave plants looking reasonably similar to the real thing (click to enlarge):

The numbers along the top give the stiffnesses, and the ones on the left give the sprout probability. The sun moves back and forth sinusoidally, which you can see in the snaking of the plants. I wasn't able to get my digital plants to spread as much as the analog one, possibly because I'm not accounting for the plant casting shadow on itself, but I'm still pleased with the results – The top center one seems particularly good. If you'd like to try for yourself, the code is here.

Tour de Living Room

A number of years ago, Marika and I got a Peloton bike, and I've often wondered whether the data from logged rides is available. This week I did some digging, and I found a couple people exploring the same question. It turns out Peloton offers the same type of REST API that I learned about when I was exploring the PUC data! Unfortunately, it's largely undocumented, but I was able to get what I was interested in by following those links above. The bike records several different statistics: the speed I'm pedaling measured in revolutions per minute (rpm), the resistance applied to the pedals measured as a percent, the output power resulting from those two factors measured in Watts, and my heart rate measured by my watch in beats per minute (bpm).

Because I'm a physicist, I was both delighted by the use of SI units for the power (Watts) and total energy (Joules), and disappointed that resistance is simply given as a percentage. I know that the output power depends on both the cadence and resistance, so we can plot those 3 together and see what the relation is:

It's a little hard to see, but if you look at points with similar resistance (color), they show a roughly linear relation between the cadence and output. As resistance increases, so does the slope. Unfortunately, the data Peloton gives is rounded to integers, so it's hard to get a precise measure of the relation.

The classes we take on the bike give target ranges for cadence and resistance over the course of a ride. The bike shows where you are relative to the min and max, and I've noticed that sometimes I can stay roughly in the center, and other times I'm ping-ponging from end to end. My impression was that it was higher resistances that made it more difficult to stay stable, so to back that up I split the rides into regions based on changes in the target resistance/cadence, and plotted the standard deviation of my cadence:

This doesn't show a relation as cut and dry as I expected, but we can see that the highest deviations all are during resistances higher than 30%, which is where I start to feel strained.

The final thing I wanted to look at was long term trends – Can I see improvement in my performance? I took some summary statistics for each ride, and plotted them with the number of days since we got it. First I looked at the distribution of output power over each ride:

The dots show the median power, and the lines show the 25th/75th percentiles. At the very beginning, I was a bit overambitious, and tried a class that was way above my abilities, then settled into a more consistent level. After a significant break (our time living in the RV, which barely has space for us and Eros, let alone a bike), I've been on a nice upward trend. The other improvement I wanted to look for was a trend in heart rates – Ideally, I should be able to achieve the same output power with a lower heart rate.

Since it takes time for my heart rate to respond to changes in effort, I decided a better measure was the max rate with the total work done over the course of each ride. You can see above, I do manage to hit higher work totals for the same heart rate, and for the lower work totals, my heart rate is lower – Progress!

If you have your own Peloton and want to see your stats (or maybe develop a Peloton analysis package?) you can find my code here.

Sync or Swim

Recently I was reminded of this demo of coupled oscillators, using metronomes on a moving platform:

Even though the metronomes are set to different frequencies and start at different points, by connecting their motion through the table, they come into sync automatically. I was curious if I could make a simple simulator that would exhibit the same effect. Each metronome has a number of torques acting to make the bob swing back and forth: there's gravity pulling down on the bob, a spring pulling the bob upright, a drag force that depends on the velocity, and an external force from the platform. In many ways this resembles the resonant pendulum I posted so long ago, but the coupling from having many oscillators makes things interesting. We can write the torque on each metronome as

The force from the platform is the sum of all the metronomes, since swinging the bob around applies a reaction force at the base of the metronome. I set up this simulation and threw in a bunch of numbers for the various parameters. After a bit of tweaking, I got something that looked pretty good:

This gets reasonably good looking by the end, but still a little out of sync. If we run it for even longer, we can look at the error between the angle of each metronome and the average angle:

It takes about 100,000 steps for them to get into a good synchronization – Not quite like the video above, but another fun toy to play with!