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3rdyearpresentation/Presentation/Presentation.tex
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@ -56,12 +56,6 @@ Washington State University \\ % Your institution for the title page
\end{frame} \end{frame}
\begin{frame}[allowframebreaks] %Allow frame breaks
\frametitle{Overview} % Table of contents slide, comment this out to remove it
\tableofcontents
%Planned TOC
% See ../outline2.txt
\end{frame}
%------------------------------------------------------------------------------------- %-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
@ -75,25 +69,105 @@ Washington State University \\ % Your institution for the title page
% %
% %
%------------------------------------------------------------------------------------- %-------------------------------------------------------------------------------------
\begin{frame}
\frametitle{Orbital Debris}
%Story from monday ISS
Recent News:
\begin{itemize}
\item Russion conducts an Anti-Satellite Missle Test generating at least 1,500 items of trackable debris
\item The Astronauts and Cosmonauts on the ISS entered lockdown, including donning pressure suits.
\item The situation is still being monitored although the immediate danger appears to have passed.
\end{itemize}
\url{https://edition.cnn.com/2021/11/15/politics/russia-anti-satellite-weapon-test-scn/index.html}
\end{frame}
%-------------------------------
\begin{frame}
\frametitle{Orbital Debris}
Other recent events highlight the dangers from orbital debris:
\begin{itemize}
\item ISS conducts 3 evasive maneuvers to doge debris in 2020-
\href{https://www.jpost.com/science/international-space-station-nearly-struck-by-chinese-satellite-debris-684809}{Jerusalem Post}
\item ISS hit by debris, May 2021- \href{https://www.asc-csa.gc.ca/eng/iss/news.asp}{Canadian Space Agency}
\item ISS dodged debris from 2007 Anti-Sat Missile, Nov 2021-
\href{https://www.jpost.com/science/international-space-station-nearly-struck-by-chinese-satellite-debris-684809}{Jerusalem Post (Same as above)}
\end{itemize}
%This isn't a unique experience
%list of other issues the ISS has faced
\end{frame}
%-------------------------------
\begin{frame}
\frametitle{Orbital Debris}
%Talk about Fregat breakup
Not just an issue for manned space flight.
\begin{itemize}
\item In May of 2020, the Satellite SL-23 Zenit Fregat's
tank suffered a second breakup event.
\item While only 65 large pieces of debris were initially identified, by Feb. 2021
over 325 had been attributed to the breakup.
\item Debris was spread in orbits between 500km and 6,000km.
\end{itemize}
\href{https://www.orbitaldebris.jsc.nasa.gov/quarterly-news/pdfs/odqnv25i1.pdf}{Orbital Debris Quarterly News - NASA}
\end{frame}
%-------------------------------
\begin{frame}
\frametitle{Why now?}
%launch costs
%nano-satellites
In recent years two major changes have occured
\begin{enumerate}
\item SpaceX: Plummeting Launch costs
\item CubeSates and other Nano-Satellites.
\end{enumerate}
\end{frame}
%-------------------------------
\begin{frame}
\frametitle{Goal of the Paper}
%Model decision making of satellite operators to be able to investigate policies to reduce kessler syndrome.
\center
Project Goals:
\begin{itemize}
\item Model the choices
facing Satellite Constellation Operators
and optimal policy policy response.
\item Investigate the effect of various policies on debris pollution
\end{itemize}
\end{frame}
%-------------------------------
%-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
\section{TOC}
%-------------------------------------------------------------------------------------
\begin{frame}[allowframebreaks] %Allow frame breaks
\frametitle{Overview} % Table of contents slide, comment this out to remove it
\tableofcontents
%Planned TOC
% See ../outline2.txt
\end{frame}
%------------------------------------------------------------------------------------- %-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
\section{Literature} \section{Literature}
%------------------------------------------------------------------------------------- %-------------------------------------------------------------------------------------
\begin{frame} \begin{frame}
\frametitle{Past Literature} \frametitle{Past Literature}
Key elements of recent literature.
\begin{enumerate} \begin{enumerate}
\item Kessler and Cour-Palais: Raised issue of orbital pollution \item \cite{Kessler1978}: Raised issue of runaway orbital pollution.
\item Adilov et al: Described 2 period salop model of interactions \item \cite{Adilov2015}: Described 2 period salop model of interactions.
\item Adilov et al: Described an infinite period model with symmetric competitive interactions. \item \cite{Adilov2018,Adilov2018a}: Described an infinite period model with symmetric competitive interactions.
\item Rao and Rondina: Describe a symetric infinite period model (first to do so). \item \cite{RaoRondina2020}: Describe a symetric infinite period model (first to do so).
\item \cite{Rao2020}: Examine the effect of Orbital-Use fees, find it would quadruple long term value produced of the space industry.
\end{enumerate} \end{enumerate}
\end{frame} \end{frame}
%------------------------------- %-------------------------------
\begin{frame}
\frametitle{}
\end{frame}
%-------------------------------
%------------------------------------------------------------------------------------- %-------------------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% Developing the Model%%%%%%%%%%%%%%%%%%%%%%%%
\section{Model} \section{Model}
@ -113,9 +187,11 @@ Washington State University \\ % Your institution for the title page
\begin{frame} \begin{frame}
\frametitle{Mathematical Notation} \frametitle{Mathematical Notation}
\begin{itemize} \begin{itemize}
\item $\{s^j_t\}$: The set of constellation satellites stocks \item $S_t$: The set of constellation satellites stocks.
\item $D_t$: The level of debris \item $s^i_t$: The number of satellites (stock) for constellation $i$.
\item $x^j_t$: The launches from each constellation \item $D_t$: The level of debris.
\item $X_t$: The set of launches.
\item $x^i_t$: The launches from constellation $i$.
\item \item
\end{itemize} \end{itemize}
\end{frame} \end{frame}
@ -129,13 +205,25 @@ Washington State University \\ % Your institution for the title page
\begin{align} \begin{align}
D_{t+1} =& (1-\delta)D_t \tag{Debris decay.} \\ D_{t+1} =& (1-\delta)D_t \tag{Debris decay.} \\
&+ g\cdot D_t \tag{Debris produced by collision with debris.} \\ &+ g\cdot D_t \tag{Debris produced by collision with debris.} \\
&+ \gamma \sum^N_{i=1} \left(1-R^i(\{s^j_t\},D_t) \right) s^i_t \tag{Debris produced by satellite destruction.} \\ &+ \gamma \sum^N_{i=1} \left(1-R^i(S_t,D_t) \right) s^i_t \tag{Debris produced by satellite destruction.} \\
&+ \Gamma \sum^n_{j=1} \{x^j_t\} \tag{Debris produced by launches.} &+ \Gamma \sum^n_{j=1} x^j_t \tag{Debris produced by launches.}
\end{align} \end{align}
\end{frame} \end{frame}
%------------------------------- %-------------------------------
\begin{frame} \begin{frame}
\frametitle{Satellite Stocks} \frametitle{Satellite Stocks}
Law of motion for satellite stocks
\begin{align}
s^i_{t+1} =& \left(
R^i(S_t,D_t,X_t)
- \eta
\right) \cdot s^i_t
+ x^i_t
\end{align}
\begin{itemize}
\item $\eta$ is the orbit decay rate of satellites.
\end{itemize}
\end{frame} \end{frame}
%------------------------------- %-------------------------------
%------------------------------------ %------------------------------------
@ -153,7 +241,6 @@ Washington State University \\ % Your institution for the title page
\end{block} \end{block}
Often described as a condition with an exponential growth of debris. Often described as a condition with an exponential growth of debris.
Similar to what is expected with Climate Change
\end{frame} \end{frame}
%------------------------------- %-------------------------------
\begin{frame} \begin{frame}
@ -163,14 +250,14 @@ Washington State University \\ % Your institution for the title page
Develops an analog of kessler syndrome where the condition is met when satellites Develops an analog of kessler syndrome where the condition is met when satellites
are destroyed immediately after launch by debris. are destroyed immediately after launch by debris.
\begin{align} \begin{align}
\left\{ (\{s^j_t\},D_t) : R^i(\{s^j_t\},D_t) = 0 \forall i\right\} \left\{ (S_t,D_t) : R^i(S_t,D_t) = 0 \forall i\right\}
\end{align} \end{align}
\item \cite{RaoRondina2020}:\\ \item \cite{RaoRondina2020}:\\
A working paper in which the authors develop a dynamic model and a definition of A working paper in which the authors develop a dynamic model and a definition of
kessler syndrome that captures all increasing debris levels. kessler syndrome that captures all increasing debris levels.
\begin{align} \begin{align}
\left\{ (\{s^j_t\},D_t) : \left\{ (S_t,D_t) :
\lim_{t\rightarrow \infty} D_{t+1}(\{s^j_t\},D_t) = \infty \lim_{t \rightarrow \infty} D_{t+1}(S_t,D_t) = \infty
\right\} \right\}
\end{align} \end{align}
\end{itemize} \end{itemize}
@ -190,7 +277,7 @@ Washington State University \\ % Your institution for the title page
\frametitle{$\epsilon$-Kessler Region} \frametitle{$\epsilon$-Kessler Region}
\begin{block}{$\epsilon$-Kessler Region} \begin{block}{$\epsilon$-Kessler Region}
\begin{align} \begin{align}
\kappa = \left\{ \left(\{s^j_t\}, D_t \right) : \forall k \geq 0, \kappa_\epsilon = \left\{ \left(S_t, D_t \right) : \forall k \geq 0,
~~ D_{t+k+1} - D_{t+k} \geq \epsilon > 0 \right\} ~~ D_{t+k+1} - D_{t+k} \geq \epsilon > 0 \right\}
\end{align} \end{align}
\end{block} \end{block}
@ -209,8 +296,9 @@ Washington State University \\ % Your institution for the title page
\begin{block}{Proto Kessler Region} \begin{block}{Proto Kessler Region}
\begin{align} \begin{align}
\kappa_\text{proto} =
\left\{ \left\{
\left(\{s^j_t\},D_t \right) : ~~ D_{t+1} - D_{t} \geq \epsilon_\text{proto} \left(S_t,D_t \right) : ~~ D_{t+1} - D_{t} \geq \epsilon_\text{proto}
\right\} \right\}
\end{align} \end{align}
\end{block} \end{block}
@ -242,12 +330,16 @@ Washington State University \\ % Your institution for the title page
%------------------------------------ %------------------------------------
\begin{frame} \begin{frame}
\frametitle{Operator's Problem} \frametitle{Operator's Problem}
\begin{align}
V^i(S_t, x^{\sim i}_t, D_t) = \max_{x^i_t} u^i(S_t, D_t) -F(x^i_t)
+ \beta \left[ V^i(S_{t+1}, x^{\sim i}_{t+1}, D_{t+1}) \right]
\end{align}
\end{frame} \end{frame}
%------------------------------- %-------------------------------
\begin{frame} \begin{frame}
\frametitle{Benefit Functions} \frametitle{Benefit Functions}
Possible profit functions Possible benefit functions
\begin{itemize} \begin{itemize}
\item Linear (Currently working on this one) \item Linear (Currently working on this one)
\item Cournot Profits \item Cournot Profits
@ -259,11 +351,26 @@ Washington State University \\ % Your institution for the title page
%------------------------------- %-------------------------------
\begin{frame} \begin{frame}
\frametitle{Planner's Problem} \frametitle{Planner's Problem}
\begin{align}
W(S_t, D_t) =& \max_{X_t} \left[
\sum^N_{i=1} \left(u^i(S_t, D_t) - F(x^i_t) \right)
+ \beta \left[ W(S_{t+1}, D_{t+1}) \right]\right] \notag \\
&\text{subject to:} \notag \\
& s^i_{t+1} = (R^i(S_t, D_t)) s^i_t +x^i_t ~~~ \forall i \notag \\
& D_{t+1} = (1-\delta + g)D_t
+ \gamma \sum^N_{i=1} \left(1-R^i(\vec s_t, D_t)\right) s^i_t
+ \Gamma \sum^N_{i=1} x^i_t
\end{align}
\end{frame} \end{frame}
%------------------------------- %-------------------------------
\begin{frame} \begin{frame}
\frametitle{Planned model expansions} \frametitle{Planned model expansions}
Multiple interacting orbital shells and debris terms. \begin{itemize}
\item Multiple interacting orbital shells and debris terms.
\item Stochastic laws of motion
\item Multiple types of operators
\item Operators benefit functions include taxation
\end{itemize}
\end{frame} \end{frame}
%------------------------------- %-------------------------------
%------------------------------------------------------------------------------------- %-------------------------------------------------------------------------------------
@ -272,11 +379,10 @@ Washington State University \\ % Your institution for the title page
%------------------------------------------------------------------------------------- %-------------------------------------------------------------------------------------
\begin{frame} \begin{frame}
\frametitle{Issues} \frametitle{Issues}
The following issues characterize the Operators' or Planner's problem The following issues characterize the Operators' and Planner's problem
\begin{itemize} \begin{itemize}
\item Curse of Dimensionality \item Curse of Dimensionality
\item Unsure of state space, no idea what equilibrium (if it exists) would look like. \item Strategic Interaction (operators only)
\item Strategic Interaction (Computability issues)
\end{itemize} \end{itemize}
\end{frame} \end{frame}
%------------------------------- %-------------------------------
@ -286,9 +392,9 @@ Washington State University \\ % Your institution for the title page
Possible approaches Possible approaches
\begin{itemize} \begin{itemize}
\item Standard VFI/Howards algorithm. \item Standard VFI/Howards algorithm.
\item VFI with sparse state space and other computational benefits. \item VFI with sparse state space (dimensionality reduction).
\item Maliar et al's approachs to supervised learning.
\item Reinforcement Learning. \item Reinforcement Learning.
\item \cite{MALIAR2018} approaches using machine learning.
\end{itemize} \end{itemize}
\end{frame} \end{frame}
%------------------------------- %-------------------------------
@ -298,9 +404,9 @@ Washington State University \\ % Your institution for the title page
% present basic approach (using my notation) % present basic approach (using my notation)
% Discuss basic training loop (use sutton and bartos term Generalized Policy Iteration) % Discuss basic training loop (use sutton and bartos term Generalized Policy Iteration)
Bellman Residual minimization \cref{MALIAR2018} Bellman Residual minimization \autocite{MALIAR2018}
Use NN to approximate $V(S_t,D_t|\theta_1)$ and $X(X_t,D_t|\theta_2)$. Use NN to approximate $V(S_t,D_t|\theta_1)$ and $X(S_t,D_t|\theta_2)$.
The loss function is: The loss function is:
\begin{align} \begin{align}
@ -319,24 +425,31 @@ Washington State University \\ % Your institution for the title page
\right] \right]
\end{align} \end{align}
Iterate between Value and Policy function estimation.
\end{frame} \end{frame}
%------------------------------- %-------------------------------
\begin{frame} \begin{frame}
\frametitle{Training Loop: Planner} \frametitle{Training Loop: Planner}
For each training epoch
\begin{enumerate}
\item Draw random data
\item train policy function
\item train value function
\end{enumerate}
\end{frame} \end{frame}
%------------------------------- %-------------------------------
\begin{frame} \begin{frame}
\frametitle{Training Loop: Operators} \frametitle{Training Loop: Operators}
Using a branched policy function and individual value funcitons
For each training epoch For each training epoch
Draw random data \begin{enumerate}
For each operator \item Draw random data
select operator's policy parameters. \item For each operator
train policy function \begin{enumerate}
train value function \item train policy function
\item train value function
\end{enumerate}
\end{enumerate}
\end{frame} \end{frame}
%------------------------------- %-------------------------------
%------------------------------------ %------------------------------------
@ -345,6 +458,9 @@ Washington State University \\ % Your institution for the title page
\begin{frame} \begin{frame}
\frametitle{State of the Code} \frametitle{State of the Code}
Currently functioning Currently functioning
\begin{itemize}
\item Nothing
\end{itemize}
%Planner training %Planner training
%Operator training %Operator training
\end{frame} \end{frame}
@ -353,6 +469,17 @@ Washington State University \\ % Your institution for the title page
\frametitle{Results} \frametitle{Results}
%simulated debris paths (increase the ammount of debris a couple orders of magnitude) %simulated debris paths (increase the ammount of debris a couple orders of magnitude)
%Protokessler region plots? %Protokessler region plots?
Results are currently waiting on finishing the code.
\vspace{12pt}
Some analyses I plan on completing include
\begin{itemize}
\item Trajectory Analysis
\item Free Entry Conditions
\item Kessler Regions
\end{itemize}
\end{frame} \end{frame}
%------------------------------- %-------------------------------
%------------------------------------------------------------------------------------- %-------------------------------------------------------------------------------------

3
3rdyearpresentation/outline2.txt
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@ -4,7 +4,8 @@ Why do we care
Possible stories: Possible stories:
sep 2019 sep 2019
2020 - 3 near misses with ISS (https://www.jpost.com/science/international-space-station-nearly-struck-by-chinese-satellite-debris-684809) (https://orbitaldebris.jsc.nasa.gov/quarterly-news/pdfs/odqnv24i3.pdf) 2020 - 3 near misses with ISS (https://www.jpost.com/science/international-space-station-nearly-struck-by-chinese-satellite-debris-684809)
(https://orbitaldebris.jsc.nasa.gov/quarterly-news/pdfs/odqnv24i3.pdf)
May 2020 - Fregat tank breakup (left debris from 1000 to 6000 miles in altitude) May 2020 - Fregat tank breakup (left debris from 1000 to 6000 miles in altitude)
May 2021 - Canadarm2 got hit (https://www.asc-csa.gc.ca/eng/iss/news.asp, part way down) May 2021 - Canadarm2 got hit (https://www.asc-csa.gc.ca/eng/iss/news.asp, part way down)
Nov 2021 - Dodge debris from anti-satellite test in 2007 (China) (Jerusalem post above) Nov 2021 - Dodge debris from anti-satellite test in 2007 (China) (Jerusalem post above)

2
CurrentWriting/sections/05_SocialPlanner.tex
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@ -5,7 +5,7 @@
The Social (Fleet) Planner's problem can be written in the bellman form as: The Social (Fleet) Planner's problem can be written in the bellman form as:
\begin{align} \begin{align}
W(S_t, D_t) =& \max_{X_t} \left[ W(S_t, D_t) =& \max_{X_t} \left[
\left(\sum^N_{i=1} u^i(S_t, D_t) - F(x^i_t) \right) \sum^N_{i=1} \left( u^i(S_t, D_t) - F(x^i_t) \right)
+ \beta \left[ W(S_{t+1}, D_{t+1}) \right]\right] \notag \\ + \beta \left[ W(S_{t+1}, D_{t+1}) \right]\right] \notag \\
&\text{subject to:} \notag \\ &\text{subject to:} \notag \\
& s^i_{t+1} = (R^i(S_t, D_t)) s^i_t +x^i_t ~~~ \forall i \notag \\ & s^i_{t+1} = (R^i(S_t, D_t)) s^i_t +x^i_t ~~~ \forall i \notag \\

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