% SIMIODETemplate-TeacherORStudent.tex and simiode.cls for LaTeX2e
% Version 29 July 2015.
% 
% This class (simiode.cls) is a modification of that used by 
% the journal PRIMUS - Problems, Resources, and Issues in 
% Mathematics Undergraduate Studies. 
% 
% See http://www.tandfonline.com/loi/upri .
% 
% The class was developed by  Marc Frantz with suggestions of the 
% PRIMUS Editorial Board, especially John Thoo.
%
% This template is to be used with the  class file "simiode.cls". 
%  
% The latest version of this class and template files 
% can be found at
% 
%       <http://www.SIMIODE.org> in the Author section.
%
% A "Quick Start Guide" appears below.
%
% This file must be processed with version LaTeX2e or higher.
%
\documentclass{simiode}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% With this version of the tex file and the associated SIMIODE.cls file you can produce BOTH a Student Version and a Teacher Version of your modeling scenario or technique narrative.
%
% Basically the Student Version contains the title STUDENT VERSION along with the NAME OF SCENARIO, with no abstract, no keywords, no tags, no author identifiers, and no COMMENTS section. The Student Version contains only the STATEMENT of the modeling scenario with appropriate figures, data, instructions, and perhaps a bibliography appropriate for student resourcing, but nothing in the bibliography which would "give away" the nature of the modeling scenario, etc.
%
%The Teacher Version contains the material in the Student Version, namely the title NAME OF SCENARIO, now headed with   TEACHER VERSION, but also full author  identification, abstract, key words, tag words, and, following the STATEMENT, a section of COMMENTS which can be discussion of presentation  ideas, "solutions," techniques, etc.
%
% Authors can prepare a full Teacher Version using the notions immediately below  to create a pdf for review (but in review please eliminate all information about the author as we use a double-blind referee system. Such information will be restored when accepted and published in SIMIODE at www.simiode.org.
%
%***********   Toggle the following line to turn the teacher version on (1) or off (0). ***********
%
%
\def\TeacherEdition{0} % Toggle this line's value.
\newcommand{\teacher}[1]{\ifnum\TeacherEdition=1 #1 \fi}
%
%
% The \teacher{ } command wraps any text that will appear only in the teacher version of
% the document. See this file for examples. This has been done for you in the document below.   
% Rename the pdf after you compile each final version.
% This option was created  by Eric Sullivan, Carroll College, Helena MT.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{url}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% NOTE regarding double spacing %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Manuscripts should be submitted doubled spaced.  This is 
% accomplished via the \baselinestretch command below.
% However, you may obtain an idea of how your article will 
% look in final form, so that you can better plan your 
% presentation.  To see your manuscript formatted very 
% similarly to the on-line published format of  SIMIODE, 
% comment the following line out by inserting a percent sign % (%) at its beginning.

\renewcommand{\baselinestretch}{1.3}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% NOTE regarding macros %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Any macros defined for your paper should be contained in 
% the top matter. Likewise, any environment definitions such 
% as \newtheorem or \newenvironment should also go in the 
% top matter. 
% 
% Do not \input your macros as separate files within your 
% final version because more files make it harder for the 
% editors and users to keep track of and/or modify and 
% append your paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Top matter is above and 
% the beginning of the document below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\PURSUIT MODELS}









\author{Brian Winkel,
Director SIMIODE\\
Chardon OH 44024 USA\\
BrianWinkel@simiode.org
}


\markboth{Pursuit Modeling}{Pursuit Modeling}





\makeStitlePDFLaTex



\begin{abstract}
Students are prompted to build systems of nonlinear differential equations to model pursuit-evader activities in which a pursuer attempts to follow, perhaps track down and come close to, an evader without knowledge of the evader's intentions, just knowledge of the evader's path history. The concept of aiming ahead of the evader's motion, in contrast to aiming directly at the position, is suggested to students.
\end{abstract}



\section*{SCENARIO DESCRIPTION}

Consider these situations:

\begin{enumerate}\item You are a driver for a {\it National Geographic\/}   photographer on the Serengeti Plain.  The photographer is gathering footage for a TV special on the cheetahs in the area. She wants to shoot footage of the cheetah chasing its prey. The real chase, as far as the driver is concerned, is getting close to the cheetah and keeping close. The prey of the cheetah is zigging and zagging and, as such, the cheetah is taking an unpredictable path so the driver needs a strategy to get close and stay close.
  \item Johnny, your ex-best friend in third grade snatched your lunch box (containing the bag of cookies your Mom made!) and he is running about the school yard, humiliating you. You want to catch him in the worst way. You know he can run about the same speed as you.  What strategy do you employ to chase him down?  It is a huge school yard with flat open fields beyond. 
  \item Your soccer team is losing 2-1. In an end-of-game time kill situation, your opponents' best skilled player on the soccer team is on the field, handling the ball very nicely.  You are determined to run him down and begin pursuing him.  What is your strategy?
  \item  You are a wolf in open tundra and the pack has retired, but you are determined to bring home the kill. You see a small deer and it sees you. Like a flash the deer starts running, seemingly in a random path. What strategy do you use to track the deer down before you get exhausted from the chase?
\item You are driving a tank in dessert warfare and  have isolated the enemy's command tank. You set out to chase down this special tank. What is your strategy?
  \end{enumerate}   

\section*{Activities}
\begin{itemize}
\item [1)] Discuss what these situations have in common and produce several strategies the individuals involved might use. Be as precise as you can with each strategy and then compare the strategies as to merits and advantages. Also be sure to define the objective of the pursuer and how to know if and when  the objective is achieved.
  \item [2)] Suppose the object being chased is called the {\it evader\/} and the object doing the chasing is called the {\it pursuer\/}. Further, suppose the position of the evader is known by the evader only, in particular NOT by the purser, and is given by the mathematical expression $(x_e(t), y_e(t))$ for time $t$ in seconds on the flat pursuit field.  Distances can be in yards. Of course even the evader (as in the case of the cheetah), may not be determining a totally independent path of motion. However, to the pursuer it appears to be just the same as if the evader purposefully designed its path.  Suppose now the pursuer's location is named (but not yet specified) as $(x_p(t), y_p(t))$ for time $t$ in seconds on the flat pursuit field. What information would the pursuer have available at and leading up to time $t = T$ seconds into the chase? How could the pursuer use this information to mount a successful chase?
\item [3)] Use a sketch to describe your strategy as a pursuer to get close to the evader.  Be sure to make your assumptions clear, e.g., what are your relative speeds?  How long can you chase at this  speed? One assumption we will offer is that the evader does not react to the pursuit. This might not be realistic. It presumes the evader is on a ``fixed'' if  weavy course, but is not responding to the pursuer.
\end{itemize}




In Figure~\ref{fig:6-8-Evader1} we see the path or trajectory of an evader and in (\ref{eq:Evader1Motion}) we see the actual parametric equations which govern the evader's motion. Recall the pursuer does not know the evader's path; just where the evader is and where the evader has been!

 \begin{figure}[!h]
\begin{center}
\includegraphics[width=10cm]{6-8-Evader1}

\parbox{9 true cm}{\caption{\label{fig:6-8-Evader1}Plot of evader's path with times of positions in seconds and distances in yards displayed. }}
\end{center}\vspace*{-.3 true cm}
\end{figure}

\vspace*{-.5 true cm}
\begin{eqnarray}\label{eq:Evader1Motion}\nonumber
xe(t)&=& 5 + \cos(2 \pi t) + 2\sin(4 \pi t)\\
ye(t) &=& 4 + 3 \cos(3 \pi  t) \,.
\end{eqnarray}



\begin{figure}[!t]
\begin{center}
\includegraphics[width=10cm]{6-8-EvaderAndPursuer1}

\parbox{11 true cm}{\caption{\label{fig:6-8-EvaderAndPursuer1} Plot of evader (thick, blue) and pursuer (thin, red) path. Here pursuer is always heading directly at where evader is at the moment.}}
\end{center}
\vspace*{-.3 true cm}

\end{figure}


 \begin{itemize}
\item [4)] Consider  the motion of the evader - the entity one is chasing,  $(x_e(t), y_e(t))$ for time $t$ in seconds. We could use the evader path  offered in (\ref{eq:Evader1Motion}) and Figure~\ref{fig:6-8-Evader1} with units yards and seconds or one of your own choosing.  Describe or model your pursuer's motion (based on some strategy you have designed), $(x_p(t), y_p(t))$, for time $t$ in seconds so that the pursuer could have a chance of achieving the goals, e.g., getting close to the evader at some instance or staying close. 

Let us ask an easier question. Can you describe  the rate of change of your motion, i.e. $(x_p'(t), y_p'(t))$ for time $t$? This is the pursuer's  speed AND direction.  How does the evader's position play a role in your pursuit strategy? How do you implement that role in any equations you might construct? NB:  Be sure you note the maximum speed ${\rm speed}(t) = \sqrt{(x_e'(t))^2 + (y_e'(t))^2}$ of your evader and give your pursuer comparable speed so as to make this challenging, but not impossible. You could also investigate the situation with various pursuer speeds. For the evader offered in (\ref{eq:Evader1Motion}) the maximum speed is 38.6 yards/second or 78.9 miles/hour.  This is realistic for a cheetah over a very short course. You might also want to examine the maximum acceleration of your evader and of your pursuer, once you construct the latter's model?  What are these in this case, i.e. what is maximum speed and maximum acceleration for evader and pursuer?

\begin{figure}[t]
\begin{center}
\includegraphics[width=9cm]{6-8-EvaderAndPursuerTracking}

\parbox{12.5 true cm}{\caption{\label{fig:EvaderAndPursuerTracking} Plot of evader (lower left) and pursuer (upper right) paths with times of positions in seconds and distances in yards displayed. The direction in which the pursuer heads at time $t = 0.2$ is directly toward the location of the evader at time $t = 0.2$.}}
\end{center}
\vspace*{-.5 true cm}
\end{figure}

  
\item [5)] With one hand trace out an evader's path on a flat surface and with  your other hand trace out a pursuer's path according to one of your strategies in ``real'' time, both at once. This is  like patting your head and rubbing your tummy! 
\item [6)] One strategy is  for the pursuer to head directly at where the evader IS at time $t$. If you have not implemented this in response to (4) then do so now and compare your previous results with the ones from this activity. What criteria do you use to compare strategy results?
\item [7)] If the pursuer heads for where the evader IS (albeit there is a moving target at the evader's location) then perhaps by the time of arrival the evader will be gone, so might the pursuer do something else?  Hint:  In basketball and soccer  one passes the ball NOT to where the intended receiver is, but to where one perceives the intended receiver will be.   Just how long is this lead distance or time?  How far should the lead be? Thus, construct a modification of activity (6) in which these considerations are used.
\item [8)] Pick an evader curve of your own. Keep it simple, like running round in a circle or running along a straight line and see what your pursuer does using your strategies. Do starting positions matter? Are there any paths for the evader with starting positions for the pursuer which give obvious pursuer paths?
   \end{itemize} 

 \teacher{
\section*{NOTES FOR TEACHERS}

This is truly an open-ended activity and can lead to a course project with many variations and nuances.  We have specifically left things open for students, but the teacher can interject at any time with some guidance.  The crucial point here is modeling the motion of the pursuer using a system of nonlinear differential equations. This can prove quite challenging, even for good students, so the teacher should be prepared to give hints and guidance.

We first learned of pursuit and evasion games from reading \cite{Arney1998}. We have worked with a number of students on projects, both in class and as independent studies and these investigations alway engage students. The real issue is modeling and attempting to implement visual and tactical strategies they conceive. This is not always easy and also the coding of this can cause some difficulty. 

We have found that offering students a good numerical solution ``package'' or computer algebra set of commands is essential so they are not wrapped up in writing detailed code for a numerical solution algorithm.  The emphasis should be on the modeling and the iterative process of ``What if we \ldots '' and ``Now I see \ldots,'' based on the feedback their technology provides. The technology should therefore permit easy changes and respond to subtle parameter value alterations. We have found {\it Mathematica\/}'s {\tt NDSolve} for numerically solving systems of nonlinear differential equations and the {\tt Animate}  feature to be very helpful, both in discovery and in  the communication of results.

The introductory notions outlined in activities (1) - (4) here are intended to get the juices going, while activity (5) calls for mathematical action and model formulation.  Activities (6) - (7) are to offer some ideas as to a straightforward pursuit strategy. Of course, this includes the natural choice that the pursuer should head directly to where the evader is. A variation on this is to head to where the pursuer thinks the evader might be, based on previous trajectory. Additional approaches include just how far ahead this lead should be taken in directing the pursuer.  We exclude all references from the Student Version, for this can be a total discovery opportunity and to give references might be to give suggestions.


There is a growing and rich literature in the area of pursuit and evader modeling. We only show the initial material which students can directly model using a system of nonlinear differential equations and a good computer algebra system, preferably one with animation capabilities. Reference \cite{ArneyPeterson2010} gives some indications as to the increasing complexity of the chase scenario and one can add multiple pursuers and multiple evaders, all the while making more and more complex the strategies of both. For example, the evader can be modeled as simply moving directly away from the nearest pursuer and the pursuer might have a sweep and corral strategy in the manner of ``go around the back of the house'' a la TV western chase situations.



This scenario, while demanding some nice geometry and calculus notions which lead to a set of non-linear differential equations, should also employ suitable technology to permit parametric equation plotting, these arising   from numerical solutions of a non-linear system of differential equations  for the pursuer's action given the evader's trajectory. We demonstrate some strategies in a {\it Mathematica\/} notebook, 6-008-T-Mma-PursuitModels-TeacherVersion.nb (and .pdf).

As quoted in \cite[p.~1419]{ArneyPeterson2010} from \cite{Isaac1965}:

\begin{quote}
Two-player pursuit and evasion games involve a single pursuer trying to catch a single evader. In the
absence of any constraints, the optimal strategies in this case are simple: the pursuer heads directly to the
evader, and the evader heads directly away from the pursuer \cite{Isaac1965}. Capture occurs if and only if
the pursuer is faster than the evader. On the other hand, if the evader's path is known precisely, the pursuer's
optimal strategy is also quite simple: head to the location where the pursuer can first catch the
evader. There are some difficulties with the assumption of full knowledge in this scenario. In his landmark
work on differential games, Isaacs goes as far as to say that any predictive tactic is a ``policy outside
the optimal strategy'', because the pursuer ``has no grounds for his prediction''\cite{Isaac1965}.
\end{quote}
\begin{thebibliography}{99}



\bibitem{Arney1998} Arney, D. C. (Ed). 1998.  {\it Military Mathematical Modeling}.  West Point NY:  Defense Automated Printing Services. Chase Problem Part 1 - pp.~109-128; Chase Problem Part II - pp.~129-140.


\bibitem{ArneyPeterson2010} Arney, C. and E. Peterson. 2010. Partial Leading in Pursuit and Evasion Games.  In  {\it Proceedings of the 2010 Winter Simulation Conference\/}.  Washington DC: IEEE. 

\bibitem{Isaac1965} Isaacs, R. 1965. {\it Differential Games\/}.  New York:  John Wiley \& Sons.  
\end{thebibliography}

}







\end{document}
