Last year from November to mid-December, a Spanish professor called Jaume visited our lab. He is a mathematician from one of the universities in Barcelona. He is interested in differential flatness, just as we control engineers are. Back in November, I wrote about a bicycle trip I did together with Jaume.
I was quite impressed when he showed me how he found a flat output for one of the systems that Susi was working on in her study thesis. The system has six states (these are x1 … x6) and three inputs (u1 … u3). It can be written in an input-affine way as:
For what I want to show you, the drift vector field f(x) does not really matter, but the input vector fields do. They are given as:
The interesting question was now how to find the flat outputs themselves. Control engineers are usually afraid of solving partial differential equations (PDEs) and therefore try to find solutions for linearized systems. But luckily, Jaume is no engineer, so he went for the PDE itself. And it turned out that he was quite lucky:
The flat outputs do not depend on x4.
The flat outputs do not depend on x6.
Well, this PDE was a bit harder. It looks like this:
But Jaume could still solve it analytically, which gave him a flat output that depended both on x3 and x5. He used the two spare flat outputs to represent both x1 and x2. Then his job was done. My job was to implement the whole thing, and it turned out that his solution worked! :-)
I wish I would be good in analytically deriving flat outputs :-/.
By the way: Whenever I write down numbers in the US, I have to keep in mind that the people here will read my German “1” as a “7”. In fact, most people around the world will only put a little serif on the top end of the vertical line instead of a huge upstroke. To find more about writing numbers, I asked Jaume to write them down in his own handwriting. So here is how a Spanish mathematician writes numbers:
The door on the picture belongs to the room of one of my floor mates. His name is “Miguel” or “Santiago” (depends on whom you ask). He is from Colombia and has just finished a language course here in Delaware.
Before I came here, he was the only floor mate which Susi had mentioned a couple of times. She calls him Miguel, just as Mary does. My two other floor mates (Iñaki and Alvaro) only call him Santiago. I don’t know why, but since they all have the same mother tongue, they must be somehow right.
Do you expect a picture of Miguel? I simply don’t have one! I’ve been in here for almost two months now, and I only saw Miguel/Santiago four times! The others don’t meet him either. He will move out in the next days (Mary told me about it, Miguel didn’t), so I probably won’t see him again. It’s so strange… there is someone who (sometimes) lives 20ft away from you but he simply does not exist…
I’ve got a new bike! The bearing in the back wheel of the old one got completely stuck right now, and getting a new back wheel would cost around $40. When I was looking through the classifieds (craigslist.com seems to be a good address!), Ji-Chul offered me a spare bike. Ji-Chul is an office mate from Korea who currently finishes his Ph.D. thesis. The bike belongs to his wife, but she has not been using it for the last few years. So here it is, the new cycle:
What is great about this bike?
- It has two brakes which both work independently of the chain!
- I can change the gears! (That’s a bit funny: I can adjust the gear changer with almost no steps, so one has to be much more cautious when changing gears compared to for example the bike I have in Germany.)
- The front suspension has a spring, so driving off the sidewalk is very comfortable now.
- It has a mounting for a bike basket. That one might become useful.
- It has a bell. Pedestrians, get out of my way!
- The handlebar is much shorter than the one of the old bike. That makes driving more sporty :-).
- And is has no fenders. That saves weight, and less weight means better acceleration!
Thank you, Ji-Chul!
However, as the bike has been rusting for quite some time, there have to be a few drawbacks. One is the second-from-top gear in the back. As the sprocket it broken, the chain spins and no torque is applied to the wheel whenever I am in that gear. So I have to get out of the 6th gear as fast as possible:
As the consulting business is really exhausting, we had to go to Miami Beach to relax a bit:
This is Christoph. He is from the “Law school” in Hamburg. And the blue thingy which he wears around his neck, that’s a slackline.Whenever Christoph finds two trees in a distance of about 6.561 yards (6m) with nothing in between, he will wrap one end of the slackline around one tree and the other end around the other tree, so that he will finally have a tight rope in between. Then he justs jumps on it and walks back and forth until he will finally fall down. This sounds easy, but as he told me, you need a really good sense of balance and a lot of training.
So, what has this to do with the “largest mystery of life”? Well, as we were having our Miami Beach sunbathes, we were drawing little parallelograms of force into the sand. We tried to figure out what force would act on the tree while Christoph was trying to keep upright. Well, while we were on the beach, we did not really figure it out. The main question was: How far is the slackline deflected while Christoph is standing on it?
While Christoph is slacklining, he has better things to do than measuring distances, so he couldn’t tell us. And we were, of course, to lazy to go to the next tree. But later that day (when it was already to dark to get sharp photos) we did the experiment and finally got our data. Here are my calculations:
From the distances you get the angles α1 = 7.59° and α2 = 18.43° which gives the following forces in the ropes:
F1 = 1513N
F2= 1581 N
where F1 is the force in the left rope and F2 is analogously the force in the right rope.
Well, that’s not that much. Sorry, Christoph, you will not chop any trees down this way, but your 7tons rope will stand it forever (it was like 7 tons, wasn’t it?) .
PS/1: I once again did not manage to include the SVG drawing into this blog post. Wp.com does not allow me to upload svg files and it seems like I cannot include files from foreign hosts. Does anyone know a way around this?
PS/2: It was easy to draw the palm trees following these instructions.
This morning I used to work on my bike once again as the chain came off twice yesterday. So … once the bike was fixed, a test drive was required. Therefore, Jaume and I decided to follow our little creek here up to Pennsylvania. (Jaume is a math professor from Barcelona who spends one month in our group here in Delaware. He currently works on a paper about differential flatness which Susi wrote during her study thesis.)
The stone you see on the first photos is a landmark between Delaware and Pennsylvania (actually, the border to Maryland is nearby, too). As you can see on the map, the border between Pennsylvania and Delaware is a 12-mile-circle while the border from Pennsylvania to Maryland is just a straight line. Did I mention that Jaume is a mathematician? *g*
Although there where many beautiful places that we went by on our way back, I only took a picture of an invisible fence. Stupid me, how can I try to photograph a fence that can’t be seen?
One last thing: Are there any good biologists/zoologists among you? Elke, Jens, Moritz? How is this fruit called? I’ve never seen it in Germany, but it seems to grow here everywhere:
PS: The bike was just great today! Although we had some high speed downhills on our way (and therefore also a couple of ascents with high torque), the chain never came off within the 2,5 hours. That is a new all-time record for me on that bike. Only once my back wheel slid away when I had to brake so hard that the wheel stalled (my bike only has a back pedal brake and no hand brakes, therefore the brake force cannot be distributed to both wheels).
Whatever problem you have, just ask Bob. He’ll fix it for you. This morning Bob took care of my bike:
The bike really is a challenge for Bob. It’s cheap. And it was probably made in China. Something falls apart all the time. In most cases, it’s just the chain. Yesterday however, it was a little screw which used to fix the fender to the frame. But Bob had another one.
Thank you, Bob!
I left Europe for eight months. This is how Europe looked like when I left it:
The first thing which attracted my attention were those baseball fields all over the place. They have them everywhere. However I have only seen these fields from the plane. In town I only went past some soccer fields with some girls playing on it. That looked really European :-).
On my flight from London Heathrow to Philadelphia I met Christoph:
We both were early for our flight so we walked around a bit on LHR airport. In fact, we both looked at the departure tables and found out that most flights heading to America would depart from the B gates. So we both took the train to the B gates (without knowing about each other). However, the B gates are boring. That’s why we went back to the A gates where all the stores are. Please note that there is no train from the B gates back to the A gates, so we had to walk. This was alright as we had plenty of time.
Christoph will be in the US for just one week. He’ll support the democratic party in Philadelphia (“Philly”) with their campaign for Senator Obama. He is paying the the flight but the Obama campaign management will compensate for his housing etc. He will have to make phone calls, he will distribute flyers and he will walk from house to house to talk to people directly.
Besides making his way into the history, he is an editor for scientific books (physics in his case). That means that he has to talk to professors and other authors about the publishing issues.
(I’m sorry, the picture is a bit blurred. On the camera display it looked alright.)