Computer Systems Lab
TJHSST
Current Projects 2004-2005
Third Period
Current Projects, 3rd Period
• Robert Brady: A Naive Treatment of Digital Sentience in a
Stochastic Game
• Blake Bryce Bredehoft: Robot World: A Evolution
Simulation
• Michael Feinberg: Computer Vision: Edge Detections
Vertical diff., Roberts, Sobels
2
Current Projects, 3rd Period
• Scott Hyndman: Agent Modeling and Optimization of a
Traffic Signal
• Greg Maslov: Machine Intelligence Walking Robot
• Eugene Mesh:
• Thomas Mildorf: A Self-Propagating Continuous
Differential System as Limiting Discrete Numerical
Construct and Device of Sorting
3
Current Projects, 3rd Period
• Carey Russell: Graphical Modeling of Atmospheric
Change
• Matthew Thompson: Genetic Algorithms and Music
• Justin Winkler: Modeling a Saturnian Moon
4
Developing a
Learning Agent
The goal of this project was to create a
learning agent for the game of bridge. I
think my current agent, which knows the
rules, plays legally, and finds some basic
good plays, is a step in the right
direction. This agent could and will be
improved upon over the course of the
year and will become smarter and learn
faster throughout the year
5
A Naive Treatment of Digital Sentience in a
Stochastic Game
Robert Brady
Abstract
My techlab project deals with the field of Artificial Intelligence or
more specifically, Machine Learning. I am designing an
agent/environment for the card game of bridge. After it learns the
rules, I will run simulations where it decides on its own what the
best play is. The level of play for the agent will increase as the year
continues because it will look up past decisions in its history to
determine what the best bid or play is in a current state of the
environment.
A Naive Treatment of Digital Sentience in a
Stochastic Game
Robert Brady
Background
Machine learning has been researched in the past and has dealt
with bridge before. This area is new, though, and anything from
intelligent agents for games to the traveling salesman problem
count as part of it. An algorithm used for one problem can be
applied in a similar manner to another such as the minimax
algorithm or the backtracking search. To build on current work,
there would have to be some sort of improvement on current
bridge-playing agents such as Bridge Baron or GIB. Both of these
programs play at a moderate level, but none of them can compare
to an expert player. The reason why an intelligent bridge-playing
agent has been hard to program in the past is that bridge is a
partially observable environment.
A Naive Treatment of Digital Sentience in a
Stochastic Game
Robert Brady
In games such as chess, or checkers, the agent could conceivably
come up with the best solution (given enough time to think about
it) because it knows where everything is. In bridge, there are
certain cards that haven't been played yet and although you may be
able to guess where they are, you can not determine this with 100%
certainty. This makes programming a learning agent for a partially
observable environment much harder. Progress For the first
semester, I worked on this program with regards to finishing
programming in the rules to the game and some simple AI
commands.
A Naive Treatment of Digital Sentience in a
Stochastic Game
Robert Brady
When this was completed about a week before the semester was
over, I began researching different AI algorithms that could be
implemented for searching. This research halted once I realized
the tree that would be searched was difficult to construct. I
consulted my professional contact Fred Gitleman and he had also
encountered this problem when programming a similar search
algorithm. He talked me through the problems I had and gave
some advice on where to find information that would help me with
those problems. During the third quarter, I worked solely on
running an algorithm (the minimax algorithm) through a depthfirst search with pruning of bad nodes.
A Naive Treatment of Digital Sentience in a
Stochastic Game
Robert Brady
The algorithm still has a few problems and will hopefully be
finished soon. Another portion of the code that I added this quarter
deals with the machine learning part of my pro ject. This part of
the pro ject stores information from the hand that the computer
just played in a file that is essentially the computer's "brain." The
brain stores information about how many tricks were taken with
the combined hands in a trump suit or in no-trump. It uses this
information for the bidding stage of the following hands. If the
numbers it reads from the file are much lower than what it believes
the current state of the environment is, it will bid higher and if the
numbers are higher, it will try to refrain from bidding.
A Naive Treatment of Digital Sentience in a
Stochastic Game
Robert Brady
My future plans include testing of the program against other
players at the school. I will use students from my tech-lab class for
a preliminary test and then after the program has established a
competitive nature with these kids, it will play against the bridge
club on Fridays during school. I hope it will be able to compete
with the students of the club at some point in the next quarter, but
if this is an unrealistic goal, I will just try and improve upon it's
algorithm as much as I possibly can before the end of the year. As
it is currently, the program needs a little more work on the
algorithm to make it fully operational before I open it up to tests
from fellow students. Co de This section is pretty much self
explanatory. Some important sections are in bold and commented
while the less important parts have been left out.
A Naive Treatment of Digital Sentience in a
Stochastic Game
Robert Brady
References
1. Fred Gitelman (fred@bridgebase.com) A programmer who also
is an expert bridge player. He advised me on how to look through a
tree to find the solution comparable to a minimax search.
2. Russell, S. and Norvig, P. Artificial Intelligence A Modern
Approach Seco Prentice Hall, NJ. 2003.
Robot Swarms
My project is an agent based
simulation, posing robots in a “game of
life”, with each new generation of robot
comes new genes using a random
number selection process creating the
mutations and evolutions that in real
life we experience for DNA
cross over and such.
13
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
Abstract:
My project is an agent based simulation, posing robots in a "game
of life", with each new generation of robot comes new genes using
a random number selection process creating the mutations and
evolutions that in real life we experience for DNA cross over and
such. There are two versions of my program a Simulation and a
Game.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
The Simulation: As stated in my abstract.
The Process: The base of my program was not the genetics, but the
graphics itself. First one robot, then a random Artificial
Intelligence for it. I then modified the world to sustain several
robots, with dying a breeding. After introducing two more Artificial
Intelligences a "group" Artificial Intelligence and a "battery"
Artificial Intelligence promoting grouping and collecting batteries
selectively. I then tweaked the environment and code until it could
sustain life. I then programed in several possible places for
environmental interaction, like viruses and the batteries. I added
the graphical output for easier analysis. Finally I created the
random number selection process to splice the genes of the parents
and create a child. The heart of the program.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
The Game:
There is the above stated "simulation" version of my program, and
a later created "game" version. The game version includes all the
same components as the simulation but also has a user controlled
robot with "laser eyes" and "grenade launchers" use to kill the
other robots. It also includes "Bosses".The Process: Taking the
simulation version and modifying it was easy, first removing the
natural deaths and graph. Then I introduced and tweaked the user
controls and the user controlled robot. Then adding lasers and
grenades and all the necessary coding, i finally added a status
display embedded window in the top left corner. I continually add
new pieces of flare to the program, such as bosses.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
1 Introduction
My project has several basic components: 3D modeling, Artificial
Intelligence, The selection process, and then basic game theory. All
these components form the amalgam that makes up my project.
Both the simulation and the game use all these components except
the simulation doesn't have any game theory.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
2 Background 2.1 Monte Carlo Simulation
Since Monte Carlo is a Simulation technique, let's first define
exactly what we mean by Simulation. A true Simulation will merely
describe a system, not optimize it! (However, it should be noted that
a true simulation may be modified in a manner such that it can be
used to significantly enhance the efficiency of a system.)
Therefore, our primary goal in Simulation is to build an
experimental model that will accurately and precisely describe the
real system. However, the breadth and extent of Simulation models
is extensive!
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
This can be illustrated by considering the three general
"classifications" of Simulation Models, below. And in each of
these "classifications", I have defined two possible
"characteristics".
1. Functional Classification Deterministic Characteristic - These
are "exact" models that will produce the same outcome each time
they are run. Stochastic Characteristic - These models include
some "randomness" that may produce a different outcome each
time it is run. This randomness forces us to make a large number
of runs to develop a "trend" in our "collection" of outcomes.
Further, the exact number of how many "runs" you must make to
obtain the "right trend" is simply a matter of statistics.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
2. Time Dependence Static Characteristic - These models are not
time-dependent. This even includes the calculation of a specific
variable after a fixed period of time. Dynamic Characteristic These models depict the change in a system over many time
intervals during the calculation process.
3. Input Data Discrete Characteristic - The input data form a
discrete frequency distribution.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
Discrete frequency distributions are characterized by the random
variable X taking on an enumerable number of values xi that each
have a corresponding frequency, or count, pi. Continuous
Characteristic - The input data can be described by a continuous
frequency distribution. Continuous frequency distributions are
characterized by a continuous analytical function of the form y =
(x) where y is defined as the frequency of x. This definition is valid
for all possible values of x (over the domain of the function).
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
We can now say that Monte Carlo Simulations are "True
Stochastic Simulations" in that they describe the "final state" of a
model by just knowing the frequency distributions of the
parameters describing the "beginning state" and the appropriate
metric that maps, or transforms, the beginning state to the final
state. They can also be either static (easy) or dynamic (more
difficult). If a prediction were required, then "every possible"
option would have to be considered and this is where the wellknown "Variance Reduction Methods" (antithetic variables,
correlated sampling, geometry splitting, source biasing, etc.) would
be used to reduce the number of iterations required in the
simulation.
Definition courtesy of JAMES F. WRIGHT, Ph.D Ltd. Co. at
http://www.drjfwright.com/
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
3 Theory
3.1 3D Modeling My graphics are done using OpenGL. In the
simulation version there are three different aspects to the graphical
output: the agents (the robots), the environment (the floor and
batteries), and the events (explosions) and the population graph.
The game version has all these same component except the agents
include a user controlled robot and bosses, the events include
grenades, lasers and grenade explosions, there is no graph, and
there is a stat indicator, and a mini map. There is also the interface
of my program from where you launch the simulations and games.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
The environment consist of a floor and background and small
cubes representing batteries. Simple enough (below right). The
robots consist of prisms and spheres to form arms, legs, torso, head
and facial features (below left). The explosions are tori spinning on
the y-axis that get more transparent as the grow in size (below
center). The is also a program that is able to output numbers for
the counters in the game version.
See appendix A.1 for example code.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
There is also a graph out put that is fairly simple. Every iteration it
plots a new point on the grid, and never erases, therefore creating a
line graph for the populations of each artificial intelligence type
(below left). The game mode utilizes a mini map and a status bar,
the bar includes life, number of grenades, and number of kills
(bottom right).
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
3.2 Artificial Intelligence There are three different main artificial
intelligences in the program and one that uses a combination of
the others. The first is a random artificial intelligence that is the
most basic, second is the group artificial intelligence that condones
forming groups for reproduction, and the last is the battery or food
artificial intelligence that promotes eating. The fourth is one that is
advanced, and has the agent use the battery artificial intelligence
when it requires energy, and uses the group artificial intelligence
when he doesn't require energy. The random artificial intelligence
first randomly decides whether to turn left, right, or go forward.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
It has a preference to turn if it turned the iteration before. This
done over time produces interesting behavior. The fact that
offspring spawn close to parents and that parents have to be close
to produce offspring means that after a while these will group, and
any robot that randomly strays from the group will die off, where
as those in the group re spawn as fast as they die.
Code is located in Appendix A.2.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
The group artificial intelligence goes through the list of robots and
first recognizes all the robots that are of a color suitable for
reproduction for the given robot. Then from this list the closest
robot is chosen, the robot then turns towards this robot and walks.
This obviously forms groups that are more efficient than the
groups produced by the random artificial intelligence, because
robots will not stray off.
Code is located in Appendix A.2.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
The battery artificial intelligence goes through the list of batteries
and finds the one closest to the robot and then turns the robot
towards it and walks. Robots will end up heading after the same
battery and form groups, these groups may or may not be able to
reproduce though, but when a pair find each other these groups
produce to be fairly strong.
Code is located in Appendix A.2.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
The group that is the strongest is an amalgamation of both group
artificial intelligent robots and battery artificial intelligent robots.
These groups stay together due to the number of group robots and
will search for food due to the battery robots. Proving to be
extremely effective in keeping alive. Sooner or later however one of
the artificial intelligences ends up getting bred out. There is a
fourth artificial intelligence that is an amalgamation of the group
artificial intelligence and the battery artificial intelligence. When
the agent is low on energy it uses the battery artificial intelligence,
until it has a decent amount then it uses the group artificial
intelligence.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
3.3 Selection Process
The selection process doesn't start with selecting but instead starts
at the beginning of every iteration. The process is began by
inventorying the population, tallying the number of robots and
their characteristics, this then produces a few tables stored as a
"genome" (code for class in Appendix A.3). This tables are formed
to make graphs of the frequency of the specific gene settings. When
the selection is called, it goes thorough and finds the optimal gene,
in the parents gene pool, using a random number process, and the
fore mentioned graphs. This is done for each gene. There is also a
level of randomness calculated in the allows for mutations. These
mutations give a new status to the gene, not in the gene pool.
Code can be seen in Appendix A.3.
Robot World: A Evolution Simulation
Blake Bryce Bredehoft
While the theory behind this selection process may seem somewhat
simple the code on the other hand is not.
3.4 Game Theory
There are lasers and grenades. Your tools for destroying the
surrounding robot population. Combine their power by shooting
the grenade as it falls with your laser and produce a powerful
explosion. After every 50 kills boss robots appear. One will appear
the first 50, two the second 50 and so on. The bosses use a boss
artificial intelligence of their own. The boss artificial intelligence
will find the user controlled robot a turn it toward it and walk.
Computer Vision:
Edge Detections
Vertical diff.,
Roberts, Sobels
33
Computer Vision: Edge Detections
Vertical diff., Roberts, Sobels
Michael Feinberg
Abstract and paper needed
Optimization of a
Traffic Signal
The purpose of this project
is to produce an intelligent
transport system (ITS) that
controls a traffic signal in
order to achieve maximum
traffic throughput at the
intersection. To produce an
accurate model of the traffic
flow, it is necessary to have
each car be an autonomous
agent with its own driving
behavior. A learning agent
will be used to optimize a
traffic signal for the traffic
of the autonomous cars.
35
Agent Modeling and Optimization of a Traffic
Signal
Scott Hyndman
Abstract
Traffic in the Washington, D.C. area is known to be some of the
worst traffic in the nation. Optimizing traffic signal changes at
intersections would help traffic on our roads flow better. This pro
ject is to produces an intelligent transport system (ITS) that
controls a traffic signal in order to achieve maximum traffic
throughput at the intersection. In order to produce an accurate
model of the traffic flow through an intersection, it is necessary to
have each car be an autonomous agent with its own driving
behavior.
Agent Modeling and Optimization of a Traffic
Signal
Scott Hyndman
The cars cannot all drive the same because all the drivers on our
roads do not drive the same. A learning agent is used to optimize a
traffic signal for the traffic of the autonomous cars. Note: The
results and conclusion pieces of the abstract are not included yet
because the pro ject is not finished.
Agent Modeling and Optimization of a Traffic
Signal
Scott Hyndman
Introduction
Traffic in the Washington, D.C. area is known to be some of the
worst traffic in the nation. optimizing traffic signal changes at
intersections would help traffic on our roads flow better. the
purpose of this pro ject is to produce an intelligent transport system
(its) that controls a traffic signal in order to achieve maximum
traffic throughput at the intersection. in order to produce an
accurate model of the traffic flow through an intersection, it is
necessary to have each car be an autonomous agent with its own
driving behavior. the cars cannot all drive the same because all the
drivers on our roads do not drive the same. a learning agent will be
used to optimize a traffic signal for the traffic of the autonomous
cars .
Agent Modeling and Optimization of a Traffic
Signal
Scott Hyndman
Background
1.1 Traffic Signal Control Strategies
There are three main traffic signal control strategies: pretimed
control, actuated control, and adaptive control.
1.1.2 Pretimed Control
Pretimed control is the most basic of the three strategies. In the
pretimed control strategy, the lights changed based on fixed time
values. The values are chosen based on data concerning previous
traffic flow through the intersection. This control strategy operates
the same no matter what the traffic volume is.
Agent Modeling and Optimization of a Traffic
Signal
Scott Hyndman
1.1.3 Actuated Control
The actuated control strategy utilizes sensors to tell where cars are
at the intersection. It then uses what it learns from the sensors to
figure out how long it should wait before changing the light colors.
For example, if the signal picks up a car coming just before the
green light is scheuled to change, the length of the green light can
be extended for the car to go through.
Agent Modeling and Optimization of a Traffic
Signal
Scott Hyndman
1.1.4 Adaptive Control
The adaptive control strategy is similar to the actuated control
strategy. It differs in that it can change more parameters than just
the light interval length. Adaptive control estimates what the
intersection will be like based on data from a long way up the road.
For example, if the signal notices that there is a lot of traffic
building up down the road during rush hour, it might lengthen the
green light intervals on the main road and shorten them on the
smaller road.
Agent Modeling and Optimization of a Traffic
Signal
Scott Hyndman
1.2 Driver Behavior
None at this time.
1.3 Machine Learning
This piece of the pro ject has not been started.
Agent Modeling and Optimization of a Traffic
Signal
Scott Hyndman
Development 2.1 Model
I am using MASON software to do my traffic simulation. MASON is
a Javabased modeling package that is distributed by George Mason
University. My simulation is based on the MAV simulation included
with the MASON download. In MASON, everything runs from the
Schedule class. The Schedule keeps track of time and moves the
simulation along one step at a time. Ob jects that move implement
the Steppable interface. Thus, each has its own Step method that the
Schedule calls at each step in time. There is also a Stoppable
interface that takes ob jects off the Schedule.
Agent Modeling and Optimization of a Traffic
Signal
Scott Hyndman
In this program, the visible simulation is made by the CarUI Car
User Interface class. The CarUI starts the CarRun class running. The
CarRun class is what starts the Schedule and creates everything in
the simulation. CarUI takes information from CarRun to display on
the screen. CarRun creates Continuous2D's, one for each of the ob
ject types used in CarRun - Car, Region, Signal, and eventController.
Continuous2D's store ob jects in a continuous 2D environment. They
make it easier keep track of the ob jects in the simulation. The
Continuous2D breaks the space of the simulation into "buckets."
Agent Modeling and Optimization of a Traffic
Signal
Scott Hyndman
If you want to find an ob ject in a certain area of the simulation, you
can check in the bucket there. For example, if you want to see if a
car has another car near its front, you can look in the bucket that that
car is in and check to see where the other cars in that bucket are. The
Car class contains the information for how each autonomous car
runs. It implements both the Steppable and Stoppable interfaces.
Regions are what goes on the background of the visual output.
Examples of Region ob jects are the roads and medians. The
Signal class is almost identical to the Region class. However, the
signals are redrawn at every time iteration while the Regions are
only redrawn if they change loaction or size.
Agent Modeling and Optimization of a Traffic
Signal
Scott Hyndman
Lastly, because there is no way in MASON to control when actions
happen in the Schedule, I made the eventController class to tell
actions when to happen. The eventController class uses functions
defined in other classes to control the ob jects of those classes.
2.2 Driver Behavior
None at this time.
2.3 Optimization
This piece of the pro ject has not been started.
Greg Maslov
Machine Intelligence
Walking Robot
47
Machine Intelligence
Walking Robot
Greg Maslov
Paper ?
Eugene Mesh
Project?
49
Eugene Mesh
Project?
Research paper?
Sorting Parts of
Variable Width
Problem Statement. To analyze the
efficacy of sort parts by using slots and
utilizing the variable angular velocities
that result when parts of distinct physical
dimensions move off of a relatively flat
inclined surface.
Purpose. The final goal is to assess the
feasibility of quality control based on
taking advantage of the different
orientations at various time after release
that are caused by deviations from the
original product.
51
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
Abstract
Apology.
In a manufacturing environment, it is crucial to establish a high
standard of quality control while at the same time maintaining a
balanced budget. Robustness of production as well as the
minimization of risk are also of concern. Ergo, simple,
automated techniques for weeding out defective pieces are
desirable. It is the intention of this pro ject to analyze the
effectiveness of one such technique.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
Purpose.
It is not uncommon for constructs to be delivered by conveyor
belts as they are processed in a factory. Their continuous motion,
when directed over the end of such a surface, induces a certain
rotation that accompanies each item during the pursuant fall.
Proposed is to sort these items by exploiting variance in this
rotation via the precise positioning of one or more slots. It is
hoped that this motion will be sufficiently sensitive to
deformation as for this procedure to be feasible.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
Scope and Procedure.
The scope of this pro ject is exploratory in nature; there is no
sense in attempting to develop a general method. Thus, we will
work with a rather simple subset of possible pieces. Moreover,
due to logistic constraints, experimentation will be conducted
primarily within a digitally rendered environment. The model
will be coded from scratch so as to give me total control of the
physics involved, and repeated trials with dependent variance
will be employed to discern the efficacy of this sorting technique.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
2 Model Synthesis
Derivation of Theoretical Equations. We will work under
admittedly simplistic circumstances, assuming that the ob jects
being sorted are approximated by a rectangular block of length,
height, and depth l, h, and d respectively. We call the generic
block B . We suppose furthermore that there exists a uniform
density within B , so that its mass M , generally given by V M=
dV is instead given by M = hld . In a real environment, products
are typically delivered by conveyor belt. Due to aforementioned
logistic constraints, we use the approximation of an inclined
plane, which we call P . B will be released from the top of P .
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
We write for the angle beween P and the gravitational
equipotential contour. Let µs and µk denote the static and kinetic
coefficients of friction, respectively, between B and P . Under our
assumptions of uniform density, the relevant calculations are
straightforward. If µs tan(), then no motion results du, to static
friction, otherwise B moves under the force of gravity and
friction. he If > - tan-1 l then the block tumbles down the
incline; we assert that this is not the case. 2 The normal
component of the contact force, Fn , is given by |Fn | = MB g
cos(), and the frictional component, Fk , by |Fk | = MB g µk
cos(). B then slides down P along path C until enough of it hangs
over the edge of P for it to begin to rotate.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
Let us call this phase of sliding initiation. Initiation is governed
by Newton's 2nd Law applied in the direction x, with the positive
direction ^pointing straight down P : F
x^
MB g sin() - MB g µk cos() = MB ax ^
= Fgx - Fk = MB ax ^ ^
ax = g (sin() - µk cos()) ^
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
1 ^ If the plane has length D in the x direction, B slides a distance
of D - 2 (l + h tan()) straight down the plane, after which it begins
to pivot about the edge of the plane. Let us call this edge e.
Calculation of its velocity v at this time can be simplified via
conservation of energy: C 1 F · dr = -P Eg - WF MB |v|2 = K E = k
2 D | 1 = MB g H - MB g µk - (l cos() + h sin()) 2 2 T D 1 v| = g - (l
cos() + h sin()) H - µk 2
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
Thus far, our equations have dealt with constants. In this phase
(angular velocity) is determined. Accordingly, we call it gliderotation. During glide-rotation, critical variables that govern the
movement of B , such as Ie and Te , the moment of interia and
torque about e respectively, are functions of the position of B
itself. Let eo denote the axis parallel to e that passes through the
3-D center of B . The calculation of the moment of intertia of B
about eo is straightforward: Ieo = V r dm =
2
2 -l 2
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
h2 + l2 h2 + l2 = MB 12 12 , h2 2 +l 2 The parallel axis theorem
yields Ie = MB where r is the distance between e and 12 + r eo .
Torque is given by Te = MB g x , where x is the horizontal
displacement of the center of the block past e. The torque
equation then gives us e T = Te = Ie = dhl MB g x = MB d dt
Where x = cos( ) r 2 l -h 2 h2 0 d x2 + y2 dz dy dx =
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
gx
h2 + l 2 12 h2 + l2 + r2 12
d dt + r2 .
A central calculation of determines the - h2 4 + sin( ) h and = +
23
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
torque due to the contact force between B and P , which in turn
yields Fn : e T = TF + TFn = Ieo k o Fn µ kh 2 ± r h2 2- 4 =
MB h2 + l2 12 g x h2 + l 2 12 T
Fn = + r2 MB g x - h2 4 µ h k2 ± r 2 1 + 12r 2 h2 + l 2
This set of implicit differential equations is much too difficult to
resolve by elementary methods, and thus requires a
computational model. he observant reader will surely notice that
the theoretical T
h µ2 + 1
k equation for Fn is asymptotic as r . This anomaly is the result
of a
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
theoretical Newtonian 2 decomposition that is invalid as the
direction of the combined contact force approaches that of r. A
graph of the normal force reveals that the interval of gravely
affected r is rather slim; hence, it will be reasonable patch this
anomaly by assuming a constant force until our equation is valid
again. Furthermore, we assume that contact between B and any
slots is by nature a rigid-body collision in which the slots are fixed
and infinitely massive. Moreover, we assume a constant elasticity E
[0, 1] for all of the collisions. Let r, po , p, i , and vi denote the
vector pointing from the center of B to the parcel of B in the
collision, a unit vector in the direction of the impulse delivered, the
impuls·e delivered, the initial angular rotation and velocity of B .
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
By our hypotheses, 1 1 1 2 2 2 E = 2 MB |vf |2 + 1 Io f . Now, 2 MB
|vi | + 2 Io i 2 r × po d = signedmagnitude Io f = i + |p|d v v 2 2+ · 1
|p| 1 |p| 1 1 2 2 E= pox + iy + poy MB |vi | + Io i MB Io (i + |p|d )2
ix + 2 2 2 MB MB 2 = 1 122 1 1 |p|2 2 0 = |p| (vi · po ) + + Io i d |p|
+ Io d |p| + (1 - E ) · MB |vi |2 + Io i 2 MB 2 2 2 ( ( 1 -(vi · po + Io i
d ) + vi · po + Io i d )2 - 2 MB + Io d2 1 - E )Ei |p| = 1 2 MB + Io d
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
1 2 where Ei is the initial energy 2 MB |vi |2 + 1 Io i and vi · po =
vix pox + viy poy . We choose + · · · 2 because as E 1- , the
expression with - · · · tends to 0. (Recall, for instance, that vi · po is
always negative.) Of course, the fixed elasticity opens the
possibility that |p| is computed to be imaginary. For such instances,
we adopt the convention of perfect elasticity, which is uniquely
determined by computing |p| with E = 1. The crux of my pro ject is
to create and refine this model to the degree that I can predict the
that will result when B is released from the top of P . Hopefully,
there will be a high enough degree of sensitivity to the dimensions
of B that I will be able to sort good and bad pieces based on the
variation in .
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
Computational Mo deling In order to separate the model itself
from the input and graphics implementation, the source has been
partitioned into three files:
1. PhysModel.cpp - The body that contains most of the physics
equations and graphics routines to render the set up.
2. Parse.cpp - The file which takes the command line input,
defaulting unassigned variables to the previous run via an
intermediary storage file.
3. Polygon.cpp - The source that defines general graphics
functions, the structs Polygon and Vertex, and several polygonal
intersection routines.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
The fundamental hypothesis of this pro ject is the assumption that
the complicated motion of the block can be modeled as a discrete
set of equations repeatedly propagated through a small time step.
The goal, then, is to apply such modeling techniques to a system
that is hopefully sufficiently complicated as to exhibit a high
degree of sensitivity to independent variables such as height and
length. Implementation begins with calls to several initialization
routines. The first call interprets the command line input.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
From the command line, independent variables can be assigned by
appending the word "*VariableName=Value" to the end of the
command line call. Graphics and a number of debugging flags can
be assigned via the word "-flags". (Graphics, for example, is the
flag "g", which then propagates as GFX=1.) Then the Sine,
Cosine, Tangent, and Sqrt look-up tables are calculated. By
precomputing the values of sine, cosine, tangent, and square root
at millions of points, we can effectively negate the costexpensiveness of computationally expense Taylor series
calculations without intolerable loss of precision. Indeed, a brief
scan of direct comparison indicates accuracy to roughly 6 correct
decimal places.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
Implementation continues with a call to display, which serves to
manage thousands of repeated calls to Model, in which the
independent variables of length, height, and the position of the
rightmost slot-block are minimally altered. The values returned are
tabulated in the file specified by the ofstream DAT. The variable
ANOMALOUS is a global that keeps track of the validity of the
computed outcome: the block being either or rejected by the slot.
The potential loss of validity is a function of the theoretical
assumptions that we have made which are not necessarily valid.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
Some potential errors and mitigating devices I have used to address
them include · Error induced by Eulerian time discretization.
Because of the complex nature of the system, it is difficult to
determine precisely how this affects accuracy. The runs I have
conducted on a typical PC use what appears to be a small time
step: between one and three hundredths of a second. · Perfectly
rigid collision. As we shall see, there is a small tolerance built into
impulse function which governs this collision. · Look-up table
error; as mentioned, calculated sufficiently many times, this error
can be reduced to on the order of 1 part in 1,000,000.
Implementation continues to the model itself.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
A few qualitative comparisons are conducted. Static friction must
meet or exceed kinetic friction, but must not be so great as to
prevent any motion at all. Moreover, the assumption that blockplane contact remains face-to-face throughout initiation asserts a
simple trigonometric relation. Finally, if the block is simply too
large to fit through the slot, the rejection is categorically recorded
as a rejection with zero anomaly. After the said preconditioning,
the model simulates the triphasic experiment in three consecutive
loops. checking its center with a set of horizontal and vertical
thresholds. Model returns 0 for a rejection and a 1 for acceptance.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
The variable IPS is employed in conjunction with SDL DELAY to
add sufficient artificial delay so as to obtain the desired number of
iterations per second. The delay function itself takes only integer
arguments; thus, all values of IPS in excess of 1000 are equivalent.
Otherwise, the loops are governed by a discrete version of the
physics described in detail above. The Collision function returns
whether or not the block overlaps with any of the slot polygons,
assigning to every edge of each shape the number of other edges it
intersects with.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
Upon return to the base loop making the call, the program calls
impulse to handle the relevant impulse transfer. impulse runs time
forwards and backwards with progressively smaller time steps until
it has precisely determined when the said collision occurred. Again,
we have employed a calculational technique to lower the number
of computations necessary while maintaining high precision. The
function then computes under the standard two-dimensional
Newtonian dichotomy:
1. Either a corner of the block has protruded over an edge of a slotblock, or
2. A corner of a slot-block has protruded over an edge of the block.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
That is, it assumes that we do not have the scenario where two
corners mutually protrude over one another. The above physics
equations are then applied. The direction of the frictional
component is determined via a sequence of vector calculations.
The normal component is then scaled according to COLFRICOF,
the fixed coefficient of friction for these impacts, and impulse is
delivered. The potential error is of the block rotating partially into
the slot-block due to its rigidity. A small fraction of the collisions
result in this anomalous motion; hence, the resultant motion is
checked by another loop governed by the Collision function. If at
least one iteration is spent with such positioning, the current
iteration is flagged with an anomaly value of 2. If too many such
iterations occur, a value of 4 is used instead.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
Finally, the acceptance / rejection of the block is determined by
checking its center with a set of horizontal and vertical thresholds.
Model returns 0 for a rejection and a 1 for acceptance.
Conclusion
In converting the records of the trials into images for a cursory
and qualitative analysis, it becomes clear that the sensitivity obeys
a semi-chaotic decay. Green pixels are plotted for trials resulting in
acceptance and black is plotted for rejection. Anomalous cases are
those appearing in red. Here we refer to the parameters of the
subsequent image.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
For this particular image, the horizontal scale is one pixel equal to
one millimeter in block variance, with the base value being plotted
in the leftmost column and increased with each pixel to right.
Vertically, one pixel is a half of a centimeter in slotwidth flux, with
the initial value at the top and increases in slotwidth with each
pixel down. The base block is one meter long and half a meter in
height. The image itself actually consists of two sensitivity tests the top half corresponds to flux in length, with the bottom
depicting sensitivity to increases in height. Small, discrete regions
of acceptance are visible.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
This exploratory experiment seems to indicate that the setup can be
manipulated to exact sensitivities of at least one part in one
hundred. Moreover, there is nothing to confine the application of
this methodology to simply one slot. Though we do not review a
precise analytic treatment here, it is readily apparent that an
additional venue of selection would be the use several ramps and
slots.
A Case Study: A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of Sorting
Thomas Mildorf.
In conclusion, further refinement in this method of sorting appears
to be possible.
Appendix A - Code
For completeness, the three source files PhysModel.cpp, Parse.cpp,
and Polygon.cpp are reproduced here. PhysMo del.cpp #include
#include #include #include #include #include #include #include
<stdlib.h> <SDL/SDL.h> <GL/glut.h> <iostream> <math.h>
<fstream> "Polygon.cpp" "Parse.cpp"
Modeling
Atmosperic
Change
My goal is to create a model of the
atmosphere over time, predicting its
strength given the increasing amount
of pollution as well as the
controversial but effective Montreal
Protocol. Many projects are in place
to save the ozone, and this model will
assist in assessing the impact of antipollution movements and determine
the longterm possible outcome given
many parameters. This model features
usercontrolled variables, allowing the
user to manipulate the year, solar flux,
and existence of anti-pollution
projects.
79
Graphical Modeling of Atmospheric Change
Carey Russell
1. Abstract The goal is to create a model of the atmosphere over
time, predicting it's strength given the increasing (and in this case,
user controlled) amount of pollutants like greenhouse gases. Many
projects are in place to save the ozone, and this model will assist in
assessing the impact of antipollution movements and determine the
long-term possible outcome giving the many and flexing
parameters.
2. Background
2.1 The Ozone
Ozone, a feared word in the media, is essential to life on Earth.
Averaging about three molecules per ten million, O3 is very rare,
representing a minute fraction of atmospheric composition. Nearly
90% of all ozone is found in the stratosphere, the atmospheric
layer between 10 and 40 km above the Earth's surface, where it
shields the surface from ultra-violet radiation (UV-B).
Graphical Modeling of Atmospheric Change
Carey Russell
Ozone filters out the high energy radiation below 0.29
mircrometers, allowing only a small amount to reach the Earth's
surface. Strongest at about 25 km altitude, this layer is known as
the ozone layer; damages to this layer result in subsequent
increases in UV-B radiation and risks of eye damage, skin cancer,
and adverse effects on marine and plant life. 2.2 The Hole In the
1980's, scientists noticed a noticeable and dramatic increase in the
amount of UV-B reaching the surface. At first, they began to
suspect and then detect a steady thinning of the ozone layer.
Scientific concern morphed into public alarm when the British
Antarctic Survey announced the detection of the first Antarctic
'hole' in 1985.
Graphical Modeling of Atmospheric Change
Carey Russell
In truth, this ozone 'hole' is not a gap in the ozone layer at all;
merely, it is a sharp decline in the stratospheric ozone concentrations
over most of Antarctica for several months during the southern
hemisphere spring. Continued research revealed and satellite data
recorded depleting ozone levels over Antarctica growing worse with
each passing year.
Graphical Modeling of Atmospheric Change
Carey Russell
1985 Antarctic Hole 2003 Antarctic Hole
http://www.ucar.edu/communications/atmosphere-timeline.html
http://www.noaanews.noaa.gov/stories/s2099.htm
Research now shows that the ozone layer over Antarctica thins to 5544% of its pre-1980s level. The result is up to a 70% deficiency for
short time periods, and at some altitudes, ozone destruction is
practically total. The picture above to the right shows the current
satellite images of the ozone hole, now more than two and a half
times the size of Europe.
Graphical Modeling of Atmospheric Change
Carey Russell
possibility that CFCs could lead to serious ozone decomposition,
policy makers worldwide signed the Montreal Protocol treaty in
1987. In brief, this protocol limits CFC production and usage. By
1992, ozone loss was continuing to increase exponentially. These
evidence prompted leaders to strengthen the Montreal Protocol. The
revision called for a complete phase out of CFC production in
industrialized countries by 1996.
http://www.eohandbook.com/ceos/part2e.html
Graphical Modeling of Atmospheric Change
Carey Russell
3.3 The Result
As a result, most CFC concentrations are decreasing around the
globe. Production in developed countries has fallen by 95%. Current
research suggest that the Montreal Protocol is working relatively
effectively. The abundance of CFCs and other ozone-depleting
substances in the lower atmosphere peaked in 1994 and has now
begun to decline. There is a key distinction, however; the rate of
atmospheric destruction is now decreasing. Many people translate
this to mean that the atmosphere is repairing itself, but
unfortunately, merely the rate of decomposition is decreasing not the
amount of decomposition itself. On the positive, resulting from the
Montreal Protocol, the ozone layer is expected to recover gradually
over the next 50 years.
Graphical Modeling of Atmospheric Change
Carey Russell
4. Materials and Apparatus 4.1 The Software (NetLogo)
NetLogo began with StarLogo. It's the next in the generation of
multi-agent model languages, building "off the functionality of [the]
major product StarLogoT, and adds significant new features and a
redesigned language and user interface." So in summary, NetLogo is
a modeling environment for simulations. It very well suited for
modeling complex system such as natural or social phenomena.
Because NetLogo is programmable, modelers can give instructions to
hundreds or thousands of independent agents operating
simultaneously. This makes it possible to discover the connection
between individual behavior and group patterns. Additionally, it lets
users open simulation and "play" with them, exploring various
conditions.
See References for citation information.
Graphical Modeling of Atmospheric Change
Carey Russell
4.2 NetLogo Details
The "Net" in NetLogo is "meant to evoke the decentralized,
interconnected nature of the phenomena you can model with
NetLogo. It also refers to HubNet, the networked participatory
simulation environment included in NetLogo. The 'Logo' part is
because NetLogo is a dialect of the Logo language.
Graphical Modeling of Atmospheric Change
Carey Russell
5. Results/Discussion
So far, the program has modeled successfully the modern day
deterioration of the atmosphere. The user is able to "switch" on or
off the Montreal Protocol and see the difference the mere existence
of the program. Additionally, the user can watch the absorption rate
and watch as the radiation reaches earth. The user is alerted when
(or if) the radiation reaches dangerous levels. The solar flux is
considered a constant, especially because on such a short time scale
(through 3500) and the small fluctuations make a mediocre
difference in the overall radiation to earth. Next on the to-do list is
continued research into the inter-workings of the atmosphere. I plan
to expand on the time scale, and allow for atmospheric regeneration,
which is expected (although not currently observable).
Graphical Modeling of Atmospheric Change
Carey Russell
Also, so far the user cannot change the amount of incoming
radiation because (as stated above) the solar flux has been defined as
constant. However, soon the ability to manipulate the IR flux will be
added. A necessary complexity in the coding is allowing for the
release of radiation emitted back from earth, which as of current is
left unattended.
6. References/Appendixes Much credit goes to Netlogo, and the
National Wildlife Association for posting so many articles
concerning the future of the ozone. The Montreal Protocol is my
guide for regeneration programs.
Genetic Algorithms
and Music
Genetic algorithms use
feedback resulting from
evaluating data sets to
optimize these data sets for
the best performance as
defined by the user. The main
data
processing is done in LISP.
The creation of audio files is
done using Csound.
90
An Investigation of Genetic Algorithms Using
Audio Output
Matthew Thompson
Abstract
This paper documents my work of researching and testing genetic
algorithms.
1 Introduction
Genetic algorithms use feedback resulting from evaluating data sets
to optimize these data sets for optimum performance, where
optimum performance is defined by the user. The main data
processing is done in LISP. The program has a simple shell script as
its frontend. CSound is used to convert the data sets to audio files,
which are heard and evaluated by the user.
An Investigation of Genetic Algorithms Using
Audio Output
Matthew Thompson
2 Research
The first area of research was into various forms of genetic
algorithms[1]. Topics covered included different methods of storing
data, such as in a tree, list, or array. In a tree, mutation operators
include subtree destructive, node swap, and subtree swap, and a
single point subtree exchange as a crossover method. In a list,
mutations can be generative, destructive, element flip, node swap, or
sequence swap, and crossover can be single point or order based. In
an array, mutations can be destructive, element flips, or element
swaps, and crossovers can be single point or variable length.
An Investigation of Genetic Algorithms Using
Audio Output
Matthew Thompson
The second area of research was into music theory, to ensure that the
program would, even with random data, produce something that
sounded decent. To do this, I wrote the program such that a melody
will stay on key, and used a hash table of notes and frequencies[2]
to accomplish this.
An Investigation of Genetic Algorithms Using
Audio Output
Matthew Thompson
3 Program Development
The initial program was made to store data in lists, mutate using
element flips, and crossover being single point. I used this type of
genetic algorithm because at the time of starting the pro ject, it was
the type of genetic algorithm with which I was most familiar. After
finishing a simple score processing function that would turn a list of
numbers into a usable audio file, I integrated this with the genetic
algorithm code so that the algorithm's evaluation function was user
input telling what the user thought of the melody a specific
population member created. Melodies were rated on a scale of 1 to
9, with higher numbers indicating a stronger like of the melody. A
shell script was written to serve as a frontend for the algorithm.
An Investigation of Genetic Algorithms Using
Audio Output
Matthew Thompson
4 Program Testing
Testing of the program involved running it over repeated trials,
using different data storage, mutation, and crossover methods, and
observing trends in the improvement of the melodies created by the
program.
An Investigation of Genetic Algorithms Using
Audio Output
Matthew Thompson
References
[1] Intro to Genetic Algorithms http://lancet.mit.edu/
mbwall/presentations/IntroToGAs/index.html
[2] Frequencies of Musical Notes http://www.phy.mtu.edu/
suits/notefreqs.html
Modeling a
Saturnian Moon
This project hopes to add to our
understanding of space systems by
providing a comprehensive simulation of
the Saturnian moon system. By doing this,
this project attempts to expose what
phenomena can't be explained with
modern models and perhaps suggest
theories to explain the unexplained.
97
Space System Modeling: Saturnian Moons
Justin Winkler
Abstract
The Saturnian moon system is home to many fascinating and
unusual astronomical phenomena. For example, Epimetheus and
Janus share orbits and exchange momentum every four years.
Hyperion has chaotic rotation. Our understanding of these
phenomena, however, is unfortunately limited. This project hopes
to add to our understanding of space systems by providing a
comprehensive simulation of the Saturnian moon system. By doing
this, this project attempts to expose what phenomena can't be
explained with modern models and perhaps suggest theories to
explain the unexplained.
Space System Modeling: Saturnian Moons
Justin Winkler
Introduction
This project focuses on the modeling of complex space systems. A
problem with the realm of modeling is that there are nearly always
discrepancies in our explanations of certain phenomena. The
purpose of this project is to create a simulation of the Saturnian
moon system in hopes of better understanding unexplained
occurrences within the system. This project therefore aims to reveal
phenomena which current models do not explain, and possibly
offer explanations of such phenomena. The scope of this project is
limited only by time and computer resources.
Space System Modeling: Saturnian Moons
Justin Winkler
By adding more parameters and factors to create more complex
and accurate models, simulations could be improved with no
foreseeable end. Unfortunately, time is limited and the calculations
necessary for such a simulation may eventually exceed the
computational resources of the lab after enough model alterations.
Nonetheless, given the current resources, this project is still able to
create a comprehensive model. Background The Saturnian moon
system is a hotbed of interesting phenomena. There are moons that
have odd orbital inclinations, there are moons that are unusually
colored, and some moons may contribute to the regulation of
Saturn's rings.
Space System Modeling: Saturnian Moons
Justin Winkler
One moon is effected by the forces within the system to such a
degree that it's rotation is chaotic. There are two moons that share
an orbit, with the appearance that one will overtake the other and
the two will collide. This does not occur, however, as every four
years they exchange momentum, making the slower moon faster
than the originally faster moon. Nowhere else in the solar system
do phenomenas such these occur in such abundance. This makes
the Saturnian moon system a natural choice for simulating.
Numerous solar system simulators exist today.
Space System Modeling: Saturnian Moons
Justin Winkler
A simple example, named Orrery, can be found at
http://orrery.unstable.cjb.net/. Other simulations have been made
concerning the N-Body problem, which attempt to find subsequent
motions of bodies based on initial parameters. One of these
simulations, which uses NetLogo, is found at
http://ccl.northwestern.edu/netlogo/models/N-Bodies. This project
will build upon these past models by applying some of their
techniques to the Saturnian moon system. One major technique
used to model space systems is Newton's Law of Universal
Gravitation. This is a basic law used in innumerable simulations.
Newton's Law of Universal Gravitation is as follows: F = (G * m1
* m2) / (r2)
Space System Modeling: Saturnian Moons
Justin Winkler
Where F is the force (newtons) exerted on a massive body through
gravity, G is the gravitational constant (approximately 6.67 × 10-11
N m2 kg-2), m1 is the mass (kilograms) of the body upon which the
gravitational force is being exerted, m2 is the mass (kilograms) of
the body that is exerting the gravitational force, and and r is the
distance (meters) between the two bodies. To incorporate this law
into a 3-D model, 3 force vectors are calculated to account for
movements in the x, y, and z directions.
Space System Modeling: Saturnian Moons
Justin Winkler
Please note that, while very important, Newton's Law of Universal
Gravitation is not the only important factor. For example, an
attempt to simulate the effects of Saturn's magnetosphere on the
may be revealing, but would have an immensely different focus.
Time is still a limiting factor in this project, after all, and the
processes that would need to be simulated for this to be an
adequate portrayal would be numerous and complex. As such, this
project will generally avoid such factors and focus on the
movement of objects within the system unless enough time can be
put aside to add these factors in. By simulating the Saturnian
moon system, one hopes to better understand the extent of our
understanding.
Space System Modeling: Saturnian Moons
Justin Winkler
By basing this simulation upon commonly used models, we can
gage how accurate and effective these models are. Furthermore,
we can determine which phenomena we know how to explain
which we don't, making it clearly which events are worth further
research. Development The simulation runs using the computer
language C. It should also be noted that OpenGL is heavily
utilized. OpenGL is a graphics library for C and C++, and was
mainly implemented for testing purposes. Because of the number
of data pieces involved for proper modeling of a space system, I
was drawn immediately to the idea of creating a struct to hold data
for each object in the system.
Space System Modeling: Saturnian Moons
Justin Winkler
/* Struct representing planets, satellites, or other massive bodies */
struct mBody { double *xLocs; double *yLocs; double *zLocs; char
*name; double mass; /* In kilograms */ double xLoc; /* X
component of location (distance from one point to the adjacent is
in kilometers) */
Space System Modeling: Saturnian Moons
Justin Winkler
double yLoc; adjacent is in kilometers) */ double zLoc; adjacent is
in kilometers) */ double xVelocity; double yVelocity; double
zVelocity; double red; double green; double blue; };
/* Y component of location (distance from one point to the /* Z
component of location (distance from one point to the /* X
component of velocity vector (km/sec) */ /* Y component of velocity
vector (km/sec) */ /* Z component of velocity vector (km/sec) */
Space System Modeling: Saturnian Moons
Justin Winkler
The pointers *xLocs, *yLocs, *zLocs are used to store previous x,
y, and z locations which are then printed to the GL window as dots,
thereby tracing the path of each body. The amount of data to be
stored is user set, with a default of 10000. It should be noted that
these pointers are used solely for graphical purposes. The string
*name stores the name of each body (ex: Titan, Saturn, Hyperion,
Mimas). This string is used to create file streams to files with their
names and the string ".txt" appended to the end (ex: Titan.txt,
Saturn.txt, Hyperion.txt, Mimas.txt). These files are then used for
data storage concerning their respective bodies.
Space System Modeling: Saturnian Moons
Justin Winkler
The variable mass refers to the objects mass. The variables xLoc,
yLoc, and zLoc all store the current location of the body (with
Saturn always at the origin). The variables xVelocity, yVelocity, and
zVelocity indicate the vector components for the speed of the object.
Red, green, and blue are used solely to determine the rgb values
with which GL prints each body to the window. This project stores
the entire space system in a single array (s[]) of this struct. The
project then iterates to the appropriate runtime, each time
recalculating the values of each mBody within the array. It should
be noted that I created helper functions that does the necessary
projectile calculations. Here follows these helper functions:
Space System Modeling: Saturnian Moons
Justin Winkler
double distance(struct mBody a, struct mBody b) { double xDist =
a.xLoc - b.xLoc; double yDist = a.yLoc - b.yLoc; double zDist =
a.zLoc - b.zLoc; return pow(xDist * xDist + yDist * yDist + zDist *
zDist, .5); } /* Recalculate an mBody's parameters during a time
step */ void recalcL(struct mBody s[], int ind) { // printf("\n\nFrom
Recalc:\nCurrent distance: %lf\n", dist); // printf("%s => x: %lf y:
%lf z: %lf\n", s[1].name, s[1].xLoc, s[1].yLoc, s[1].zLoc);
Space System Modeling: Saturnian Moons
Justin Winkler
// printf("%s => xVel: %lf yVel: %lf zVel: %lf\n\n", s[1].name,
s[1].xVelocity, s[1]. yVelocity, s[1].zVelocity); s[ind].xLoc =
s[ind].xLoc + (s[ind].xVelocity * tstep); s[ind].yLoc = s[ind].yLoc +
(s[ind].yVelocity * tstep); s[ind].zLoc = s[ind].zLoc +
(s[ind].zVelocity * tstep); } void recalcV(struct mBody s[], int ind,
int numbodies) { double g = 6.6742 * pow(10, -20); double xDist;
double yDist; double zDist; double dist; double gforce = 0; double
xForce = 0; double yForce = 0; double zForce = 0; int number;
Space System Modeling: Saturnian Moons
Justin Winkler
for(number = 0; number < numbodies; number++) { if(number !=
ind) { xDist = s[number].xLoc - s[ind].xLoc; yDist =
s[number].yLoc - s[ind].yLoc; zDist = s[number].zLoc s[ind].zLoc; dist = pow(xDist * xDist + yDist * yDist + zDist * zDist,
.5); gforce = ((g * s[number].mass * s[ind].mass) / (dist * dist));
xForce += gforce * (xDist / dist); yForce += gforce * (yDist / dist);
zForce += gforce * (zDist / dist); } } // printf("\n\nFrom
Recalc:\nCurrent distance: %lf\n", dist); // printf("%s => x: %lf y:
%lf z: %lf\n", s[1].name, s[1].xLoc, s[1].yLoc, s[1].zLoc); //
printf("%s => xVel: %lf yVel: %lf zVel: %lf\n\n", s[1].name,
s[1].xVelocity, s[1]. yVelocity, s[1].zVelocity); s[ind].xVelocity =
s[ind].xVelocity + ((xForce * tstep) / (s[ind].mass)); /* Newton's
Gravitational Constant */
Space System Modeling: Saturnian Moons
Justin Winkler
s[ind].yVelocity = s[ind].yVelocity + ((yForce * tstep) /
(s[ind].mass)); s[ind].zVelocity = s[ind].zVelocity + ((zForce * tstep)
/ (s[ind].mass)); } First of all, please note that commented sections,
particularly printfs, are most likely used for debugging. As for the
functions, distance is pretty self-explanatory, returning the
distance between mBody a and mBody b. recalcV recalculates the
velocity of the mBody at the passed index (ind) by summing the
gravitational force vertors exerted by surrounding bodies and
recalcL recalculates the velocity of the mBody at the passed index
(ind) based upon the current velocity.
Space System Modeling: Saturnian Moons
Justin Winkler
These commands are executed for each iteration of the program
for every mBody. Note that tstep is a global variable that denotes
the amount of time that passes in an iteration, and the size of tstep
determines the accuracy of the model (accuracy is indirectly
related to tstep). Tstep can be user set, with a default of 500
seconds. These sections of code are primarily responsible for all
calculations (excluding the for loop that calls them). After I
created these functions, I focused on making the project easier to
test by setting up a graphical depiction using OpenGL, as well as
setting up filestreams to store data.
Space System Modeling: Saturnian Moons
Justin Winkler
As such, these calculator functions have changed very little since
their conception. I believe that now I will focus again on
calculations. Here are some ideas I intend to implement in my
code: -First and foremost, I need to get a plotting software to allow
for proper analysis of the stored data. I am currently tinkering with
gnuplot. -Another major concern is that I need to account for the
irregular shape of the bodies, particularly concerning moons or
objects similar to Hyperion (the irregular shape of Hyperion may
partially account for its chaotic motion). This will be a difficult
idea to implement, but I believe it is necessary for the accuracy of
the model.
Space System Modeling: Saturnian Moons
Justin Winkler
I believe I can accomplish this by having each body being
composed of numerous particles, and each will be acted upon by
gravity. The mechanics of this technique, however, require more
research. -Because of the growing computational demands of this
program, I have been considering using MPI to increase the
programs speed. This task should be relatively simple to
accomplish. -For an accurate model, I need accurate data on the
initial and relative positions of the bodies in question. I will need to
obtain a sky chart for Saturn (or something to that effect). -There
are many more phenomenas that could be modeled, and I intend to
look into the simulation of these phenomenas at a later date.