by admin on Nov.10, 2012, under Uncategorized
A few months ago my roommate won $500, but he only had to enter $20 to play the game. It was the first time he played, and so naturally all of the regulars were very upset. The game works like this: A 10×10 table is set up, with each row and each column corresponding to a digit 0-9. These digits represent the final digit in the score of the home team and the away team. One quarter of the money goes to the person who bought the square at the intersection of the last digits of the two team’s scores in the first half. Another quarter is paid out based on points only scored in the second half. The final half of the money goes to the person whose name was in the box which corresponds to the last digits of the final scores of the two teams.
If football scores were random, you could pick any box, cross your fingers, and have an expected payout of exactly 1 (no taxes or rake). But scores aren’t random. They are only added in discrete amount of 2, 3, 6, 7, and 8. Some boxes had to be better than others, and I needed to find out exactly how much better they can be.
By using the statistics at http://www.pro-football-reference.com and a combination of PERL scripts and Excel, I was able to come up with some neat conclusions. The data I used was from every pro game (including playoffs) from the 1974 season through November 10th, 2012. This is a huge dataset that includes over 18,000 final scores.
First I looked at overall scores. I was expecting to see some scores happen more frequently than others, but I was surprised at how extreme the effect was.
The highest peak is a score of 17, which is seen in almost 8% of final scores. Nearby peaks are 10, 20, and 24, all at about 6.5%. The double peak of 13 and 14 is also not a surprise. But I still hadn’t answered my question. I didn’t need to know about the final score, just the last digit of the final score. So I ran the same analysis across all 18,000 final scores to get a similar chart.
If the final digit were random you would expect the results to all be hovering around 10%. Obviously there is something else going on here because you can see the same oscillations from the first chart again in this chart. It appears that 0, 3, 4 and 7 occur more frequently than you would expect, and might give you an edge when you are betting. But how much of an edge? It gets more complicated when you combine a great number, like 7, with an average number, like 6. Is it still worth it? How worth it? To answer these questions I compiled a cheat sheet showing the expected return on investment for each of the 100 positions on the chart (note that the chart is mirrored across the diagonal because it shouldn’t matter which team is which).
So now we know that betting on 7 and 7 is the best one to choose, and betting on 2 and 2 is a good way to throw away your money. But in this game you can place multiple bets on the same board. How does that change the betting strategy? If you rank each box by expected value and keep betting on more and more boxes then you should expected your probability of winning to rise while your expected value shrinks, until they both equal one (when you bet on every square and win all of your own money back). This last chart illustrates the entire betting spectrum.
When making this chart I was also able to demonstrate (accidentally) the error in computing experienced when multiplying together 100 different numbers that are all very near – but below – one. Both curves should be approaching 1 or 100% as the number of bets approaches 100. However truncation/rounding error causes both curves to miss their final target (both the probability of winning and the expected value are reported too low in all cases). Because of the way I calculated the probabilities, I would expect the accuracy to decrease as the number of bets placed increases. A linear scaling (probably not the right way to take the errors into account) of the previous chart results in a different chart.
The initial iteration of the auger was just installed into the nearly completed structure. Mike made some pretty creative parts that will be assembled in the near future.
These clips hold the door brush to the side of the PVC sheel around the auger. The brush is critical because without the resistance of the bristles, the balls would just roll down the auger. The clip on the left is different because it needs to clear the motors on the agitation assembly.
These two waterjet parts were, as usual, provided by the generous Richie P. The one on the left is where the 30 tubes of balls will plug into the agitation/distribution assembly. The circle on the right is made of steel instead of aluminum. It acts like a ring gear, and with rubber triangles along the perimeter, is driven to disrupt any balls that might be jammed at the entrance to one of the 30 holes.
This video highlights the first time we turned on the auger after it was mounted in the main structure. Besides showing the coolness of the auger, it also shows all the progress we made on the structure!
The main structure was assembled and is now able to support all of the keys, as well as a big bucket of balls.
The sloped basin which funnels the balls into the auger:
Raised key units with some main structure:
One of the wings:
For the second assignment in KDC the group needed to control a spaceship and land it on a free rotating Alien Artifact. This was particularly difficult because the artifact didn’t have uniform density, so it would spin seemingly randomly in 3-space. The only feedback we had on the artifact was the relative location of its 8 corners. After observing it briefly, we were able to determine the principal axis of inertial and predict its motion using forward dynamics. Here is a video of our lander approaching and docking with the artifact. Note that while it looks like we just “stick” to the artifact, we are actually using thrusters to stay perfectly adjacent to it without ever making contact.
After a bit of confusion with OnlineMetals.com, I got all 94 feet of Aluminum 1”x1” in the mail. I cut it up into the appropriate lengths using the cool new carbide-tipped miter saw in the robotics club. In the end, there are 54 lengths of angle that all need holes drilled in specific places. All these pictures are from my cell phone, so bear with me…
40 of the 54 lengths needed for the vibraphone:
Key Units Painted
The first attempt at painting the key units was a complete failure. However, Megan Dority suggested that I used primer, and even picked some up for me. It worked like a charm. Also, Austin Buchan was able to get the group access to the Newel Simon paint booth, which was a huge help.
Key units drying in the paint booth:
Key units after drying:
Waterjet Parts Donated
RobOrchestra founder and alumnus Rich Pantaleo came through once again for the group. He was able to obtain a large donated sheet of 6063 Aluminum in just a few days when we needed it most. Also, he cut out all the parts we needed just a few days after getting the sheet! I made a few mistakes on those parts, but nothing too critical. I had to widen half of the slots in the giant half circles because I didn’t account for the thickness of the paint and the oversized hardboard.
Completely assembled and painted key unit:
All of the key units on their half circles:
After a successful design review at the weekly Wednesday meetings, the final modifications were made to the Vibratron design. After only a few major changes, the completed Vibraphone design looks something like this:
Instead of relying on tension in cables or cloth to keep the wings in their proper place, kickstands were added to each wing to keep it in the right position. The kickstands also serve as the mounts for the cables that will be keeping the cloth tensioned.
All surface that could potentially come into contact with the steel balls are covered in a 1/8” thick layer of neoprene foam. The foam will be attached with an adhesive instead of using hardware. The longest diagonal of the final outer area of the robot are just under 8 feet. Despite a few minor edits in the basin, the addition of the kickstands, and some other tweaks, Vibratron is still able to fold up into a neat 1’x1’x4.5’ column for storage and transport (excluding the two separate racks of key units).
The entire structure is made out of 94 feet of aluminum 1”x1” angle, varying in thickness from 1/16” to 1/4”. That aluminum has been ordered ($140.61) and fabrication of the main structure should be underway before mid-February.
With only $200 left in the $1,000 budget, the group still needs a 48’x36”x1/4” sheet of aluminum to waterjet into some very important pieces. Using cheap 3003 H-14 aluminum sheet, it will cost $160 just for the raw materials for those pieces. That leaves only $40 in the budget for fabric, foam, a power supply, steel cables, and other hardware. Obviously the ends won’t be meeting, so we need to look for a donation of the aluminum plate.
Kinematics, Dynamic Systems, and Controls (16-711) is one of the most interesting and challenging classes I have ever taken. We had a very difficult first assignment that involved controlling a simulated 2 link 2D robot arm. Here is a video just to demonstrate an idea of what was going on:
At every timestep in the simulation, we are given the current joint angles and velocities, and we need to return values for torques that we want to apply at each of the two joints to control the arm. Basically, we used PID Control to do some really cool stuff, including having the robot write my signature!
Jon was an incredible partner on this assignment and I look forward to working with him again. You can view our final submission here. I can’t wait till the next project!
Something I never considered when designing previous versions of Vibratron was its ability to fit through doors. While the old design could fit through a set of double doors, we wanted the entire robot to be able to fit through a standard door. This change in criteria required some major changes in the design of a few parts of the Vibraphone. It also allowed for a few other system upgrades in the process. Keep in mind that all of the renders here do not include any foam/cloth skin that will be used to contain the balls.
Giant Circle Full of Key Units
The large waterjet circle that held up all of the key units was three feet in diameter. Combined with the overhang of some of the key units, the diameter of the robot was at nearly five feet. Separating the giant circle into two large semicircles fixes the problem pretty easily. Hand grips were added so that the semicircles could be carried easily. Even though they are 25 pounds each, the semicircles can be carried close to the body with arms locked, which is a requirement for simple transportation of the machine.
Fold Out Wings
Instead of the large fixed upside-down-umbrella style from the previous design, this design has four fold out “wings” that catch the balls and funnel them towards the center. The overall diameter is six feet when open, but the wings can fold up completely vertically alongside the column. Between each wing is a pie-wedge shaped piece of cloth or foam. This has a duel purpose of funneling balls toward the center and regulating the deployment height of the wings. When the wings are raised, the compliance of the cloth/foam will allow it to fold.
Deeper Square Basin
The previous basin was a thin circle, but our research with the prototype of the recirculation system has suggested that we will need many more balls in the system to reach steady state. A wide square basin rigidly integrated with the vertical columns can hold the necessary volume of balls. Four trapezoidal sheets of plastic also keep the balls rolling towards the center of the basin.
After several attempts at getting the 180 separate parts necessary for the previous vibratron key unit laser-cut, we finally found a feasible method for fabrication. The father of a roboclub member offered to us the use of his large CNC routing table. Because Acrylic does not machine well (it is much too brittle) some redesign was done to make the key units out of hardboard.
The biggest change between designs was the decision to not remove the material between key points, exchanging concave cutouts for straight lines. Each new unit is made of five separate pieces of hardboard, connected by wood glue (instead of plastic welding). Only two parts per unit are unique, instead of 3, which makes machining prep and assembly easier.
Circular Structure with New Key Units
The new key units attach to a horizontal 1/4” plate, just like the previous version. The only difference is that instead of two clips and two colder pins, these units attach with just a colder pin. Nothing else in the structure needed to be modified to accommodate the change.
All of the pieces for all 30 key units can fit on five sheets of 2’x4’ hardboard. Hopefully these items will all be machined by the end of the winter break so focus can be shifted to the design and fabrication of the structure instead. Below is an example of how the pieces fit on a sheet of hardboard. The labels are engraved .02” into the board, and everything else is a profiling cut.
I was able to complete the game of Simon that I was developing for my cousins in time. The hardest part by far was getting the IR sensors to cooperate. Here are some photos of the final product, followed by a really artistic video my roommate Mike made for me.