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Title: Computer Systems Lab TJHSST Current Projects 2004-2005 Third Period


1
Computer Systems LabTJHSSTCurrent Projects
2004-2005Third Period
2
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
    DetectionsVertical diff., Roberts, Sobels

2
3
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
4
Current Projects, 3rd Period
  • Carey Russell Graphical Modeling of Atmospheric
    Change
  • Matthew Thompson Genetic Algorithms and Music
  • Justin Winkler Modeling a Saturnian Moon

4
5
Developing a Learning AgentThe 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
6
A Naive Treatment of Digital Sentience in a
Stochastic GameRobert 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.

7
A Naive Treatment of Digital Sentience in a
Stochastic GameRobert 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.

8
A Naive Treatment of Digital Sentience in a
Stochastic GameRobert 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.

9
A Naive Treatment of Digital Sentience in a
Stochastic GameRobert 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 depth-first search with
    pruning of bad nodes.

10
A Naive Treatment of Digital Sentience in a
Stochastic GameRobert 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.

11
A Naive Treatment of Digital Sentience in a
Stochastic GameRobert 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.

12
A Naive Treatment of Digital Sentience in a
Stochastic GameRobert Brady
  • References
  • 1. Fred Gitelman (fred_at_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.

13
Robot SwarmsMy 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 DNAcross over and such.
13
14
Robot World A Evolution SimulationBlake 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.

15
Robot World A Evolution SimulationBlake 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.

16
Robot World A Evolution SimulationBlake 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.

17
Robot World A Evolution SimulationBlake 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.

18
Robot World A Evolution SimulationBlake 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!

19
Robot World A Evolution SimulationBlake 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.

20
Robot World A Evolution SimulationBlake 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.

21
Robot World A Evolution SimulationBlake 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).

22
Robot World A Evolution SimulationBlake 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 well-known
    "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/

23
Robot World A Evolution SimulationBlake 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.

24
Robot World A Evolution SimulationBlake 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.

25
Robot World A Evolution SimulationBlake 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).

26
Robot World A Evolution SimulationBlake 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.

27
Robot World A Evolution SimulationBlake 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.

28
Robot World A Evolution SimulationBlake 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.

29
Robot World A Evolution SimulationBlake 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.

30
Robot World A Evolution SimulationBlake 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.

31
Robot World A Evolution SimulationBlake 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.

32
Robot World A Evolution SimulationBlake 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.

33
Computer Vision Edge DetectionsVertical diff.,
Roberts, Sobels
33
34
Computer Vision Edge DetectionsVertical diff.,
Roberts, SobelsMichael Feinberg
  • Abstract and paper needed

35
Optimization of a Traffic SignalThe 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
36
Agent Modeling and Optimization of a Traffic
SignalScott 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.

37
Agent Modeling and Optimization of a Traffic
SignalScott 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.

38
Agent Modeling and Optimization of a Traffic
SignalScott 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 .

39
Agent Modeling and Optimization of a Traffic
SignalScott 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.

40
Agent Modeling and Optimization of a Traffic
SignalScott 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.

41
Agent Modeling and Optimization of a Traffic
SignalScott 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.

42
Agent Modeling and Optimization of a Traffic
SignalScott Hyndman
  • 1.2 Driver Behavior
  • None at this time.
  • 1.3 Machine Learning
  • This piece of the pro ject has not been started.

43
Agent Modeling and Optimization of a Traffic
SignalScott 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.

44
Agent Modeling and Optimization of a Traffic
SignalScott 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."

45
Agent Modeling and Optimization of a Traffic
SignalScott 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.

46
Agent Modeling and Optimization of a Traffic
SignalScott 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.

47
Greg MaslovMachine IntelligenceWalking Robot
47
48
Machine IntelligenceWalking RobotGreg Maslov
  • Paper ?

49
Eugene MeshProject?
49
50
Eugene MeshProject?
  • Research paper?

51
Sorting Parts of Variable WidthProblem
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
52
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

53
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

54
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

55
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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 .

56
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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 gt - 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.

57
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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())

58
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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 v2 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

59
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
Mildorf.
  • Thus far, our equations have dealt with
    constants. In this phase (angular velocity) is
    determined. Accordingly, we call it
    glide-rotation. 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

60
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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

61
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
Mildorf.
  • g x
  • h2 l 2 12 h2 l2 r2 12
  • d dt r2 .
  • A central calculation of determines the - h2 4
    sin( ) h and 2 3

62
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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

63
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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 impulse delivered, the initial angular
    rotation and velocity of B .

64
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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 pd v v 2 2 1 p
    1 p 1 1 2 2 E pox iy poy MB vi Io i
    MB Io (i pd )2 ix 2 2 2 MB MB 2 1 122 1 1
    p2 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

65
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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 .

66
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

67
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

68
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
Mildorf.
  • From the command line, independent variables can
    be assigned by appending the word
    "VariableNameValue" 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 GFX1.) 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 cost-expensiveness
    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.

69
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

70
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

71
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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
    block-plane 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.

72
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

73
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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 slot-block, or
  • 2. A corner of a slot-block has protruded over an
    edge of the block.

74
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

75
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

76
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

77
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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.

78
A Case Study A Self-Propagating Continuous
Differential System as Limiting Discrete
Numerical Construct and Device of SortingThomas
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 ltstdlib.hgt ltSDL/SDL.hgt ltGL/glut.hgt
    ltiostreamgt ltmath.hgt ltfstreamgt "Polygon.cpp"
    "Parse.cpp"

79
Modeling Atmosperic ChangeMy 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 anti-pollution
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
80
Graphical Modeling of Atmospheric ChangeCarey
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).

81
Graphical Modeling of Atmospheric ChangeCarey
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.

82
Graphical Modeling of Atmospheric ChangeCarey
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.

83
Graphical Modeling of Atmospheric ChangeCarey
Russell
  • 1985 Antarctic Hole 2003 Antarctic Hole
  • http//www.ucar.edu/communications/atmosphere-time
    line.html http//www.noaanews.noaa.gov/stories/s20
    99.htm
  • Research now shows that the ozone layer over
    Antarctica thins to 55-44 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.

84
Graphical Modeling of Atmospheric ChangeCarey
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

85
Graphical Modeling of Atmospheric ChangeCarey
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.

86
Graphical Modeling of Atmospheric ChangeCarey
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.

87
Graphical Modeling of Atmospheric ChangeCarey
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.

88
Graphical Modeling of Atmospheric ChangeCarey
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).

89
Graphical Modeling of Atmospheric ChangeCarey
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.

90
Genetic Algorithms and MusicGenetic algorithms
use feedback resulting from evaluating data sets
to optimize these data sets for the best
performance as defined by the user. The main
dataprocessing is done in LISP. The creation of
audio files is done using Csound.
90
91
An Investigation of Genetic Algorithms Using
Audio OutputMatthew 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.

92
An Investigation of Genetic Algorithms Using
Audio OutputMatthew Thompson
  • 2 Research
  • The first area of research was into various forms
    of genetic algorithms1. 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.

93
An Investigation of Genetic Algorithms Using
Audio OutputMatthew 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 frequencies2 to accomplish this.

94
An Investigation of Genetic Algorithms Using
Audio OutputMatthew 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.

95
An Investigation of Genetic Algorithms Using
Audio OutputMatthew 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.

96
An Investigation of Genetic Algorithms Using
Audio OutputMatthew Thompson
  • References
  • 1 Intro to Genetic Algorithms
    http//lancet.mit.edu/ mbwall/presentations/IntroT
    oGAs/index.html
  • 2 Frequencies of Musical Notes
    http//www.phy.mtu.edu/ suits/notefreqs.html

97
Modeling a Saturnian MoonThis 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
98
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.

99
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.

100
Space System Modeling Saturnian Moons Justin
Winkler
  • By adding more para
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