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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Drone Package Delivery}





\markboth{Drone Package Delivery}{Drone Package Delivery}

\author{Robert Krueger\\
Concordia University, Saint Paul\\
St. Paul  MN  55104 USA\\
rkrueger@csp.edu
\and
Eric Stachura\\
Kennesaw State University\\
Kennesaw, GA 30144 USA\\
estachur@kennesaw.edu
}


\makeStitlePDFLaTex


\begin{abstract}
In this scenario, students will begin by carefully reading through the problem statement and uncovering which information is useful. Students will derive a system of first order differential equations which describe the flight path of a drone delivering a package. Techniques used to derive the analytical solution include similar triangles via one method, and vector analysis with another method. We suggest an emphasis on numerical methods as well, and have included resources on the Euler method, Improved Euler method, and Runge-Kutta methods for the numerical analysis of the solution. Finally, we have included a beamer file to introduce this problem in class.
\end{abstract}


\section*{SCENARIO DESCRIPTION}

You just started a new job with Amazon\texttrademark! As a newbie, you are  to practice delivering packages to houses with  drones. So, you start with your own house! Your friend Jimbo volunteers to take the drone 5 miles east of your home, with the goal of flying it directly back to your home. The drone is 12.6 $\times$ 12.6 $\times$ 4.2 (all in inches) in size. You program the black drone to always head directly to your house at a speed of \textbf{b} mph. Your house is located at the corner of 8$^{\rm th}$ Ave. and 19$^{\rm th}$ St. in New York City. There is a wind blowing from the south at a steady \textbf{w} mph.  Assume the drone always flies at the same height. What is the flight path of the drone? Can you use numerical analysis to estimate the solution trajectory? What methods can you use? 

%The convention in naming scenarios - x-y-NoSpacesInName-StudentVersion or x-y-NoSpacesInName-TeacherVersion - is found in Table~\ref{table:conv}. The x-y designators will be added by the Editor upon acceptance.
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%Teacher & x-y-NameScenario-TeacherVersion & x-y- stands for topic or unit (x) and Number in that topic or unit (y).\\ \hline
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%Student & x-y-NameScenario-Data-StudentVersion & Here Data means this is a data file, in this case for Student, otherwise for Teacher\\ \hline
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%\begin{itemize}
%\item [] Modeling One  -  First Order Differential Equations 
%\item [] Modeling OneA -  Numerically Solving Differential Equations 
%\item [] Modeling Two - Limited Growth Models 
%\item [] Modeling Three - Second Order Homogeneous Differential Equation Models 
%\item [] Modeling Four - Second Order NonHomogeneous Differential Equation Models 
%\item [] Modeling Five - Linear Systems of Differential Equation Models 
%\item [] Modeling Six - NonLinear Systems of Differential Equation Models 
%\item [] Modeling Seven - Changing the Venue for Solution Success - Laplace Transforms 
%\item [] Modeling Eight - Representing Natural Phenomena with Sines and Cosines 
%\item [] Modeling Nine - Modeling with Differential Equations in Higher Dimensions
%\end{itemize}


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\teacher{
\section*{NOTES FOR TEACHERS}
At the national Science Foundation funded SIMIODE MINDE - Model INstructors in Differential Equations Workshop - workshop at Manhattan College, the project 6-023-Drone Heading Home was discussed by the authors. We have adjusted the problem statement from \cite{mainreference}. There were a few goals in doing this Modeling Scenario:%The following are comments based on feedback of other participants as well as the second named author's experience using this in class. 
%We have included a variant of the problem statement of \cite{mainreference} in Section \ref{sec:ProblemStatement}; there were a number of pedagogical reasons for this. Section \ref{sec:PastUse} discusses the second author's implementation of this project in a first course in differential equations. Then in Section \ref{sec:LessonPlan} we suggest a lesson plan to discuss this problem in class, based on the authors' presentation of this work at the MINDE workshop. In Section \ref{Sec:Vectors} we give an alternate solution to the main problem, using vectors instead of geometry and trigonometry. Finally, in Section \ref{Sec:Analysis} we discuss numerical analysis of the differential equation. In particular, we provide various numerical methods, and include associated Mathematica notebooks, demonstrating the use of Euler's methods and Runge-Kutta methods to solve the main differential equation. We expect the more sophisticated numerical methods can be adjusted for use as a student project in a first course in differential equations, especially since Runge-Kutta methods are not always covered in such a course. 




\begin{itemize}
\item To have the students pick out any keywords from the prompt; 
\item To have the students discard any erroneous information; 
\item To define any relevant variables.
\end{itemize}
We have also included a brief calculus review in the Appendix, which can be printed out and handed to the students in class when introducing the drone problem. Any calculus textbook will have the necessary material in the textbook, but we have included a suggested review of the relevant concepts useful for this project.  

\subsection*{Past Use}
The second named author used the original \cite{mainreference} in a first course in ordinary differential equations taught at Haverford College in the Spring of 2018. This problem was assigned as a partner writing project to be completed outside of class. The problem was not discussed at all within class. %Section \ref{sec:LessonPlan} provides a sample lesson plan to introduce this problem within one class period, and have the students complete the writing project outside of class. 

The structure of the writing project was as follows: the students were given two weeks to complete a draft model, which the instructor would check for accuracy. Then the students would have one week to follow up on any revisions to the model, and complete the final draft of the paper. Some common challenges the students faced in this project included:

\begin{itemize}
\item Remembering parametric curves from calculus;
\item Differentiating between scalars and vectors when analyzing velocity and speed; 
\item Taking a snapshot in time with no wind (this seems to have been the largest source of confusion).
\end{itemize}


\subsection*{Possible Lesson Plan} %\label{sec:LessonPlan}
\noindent A sample lesson plan for one 50 minute class introducing the problem is as follows, with a useful vector calculus review in the Appendix. We have also included a beamer file that can be used during the class period to introduce the problem. Finally, in boldface with black font are suggested times for activities, while in boldface with underlined blue font are suggested hints to provide along the way (see the beamer file). 


\begin{description}
\item[10 minutes] Have the students pair off and read the problem carefully.
\end{description}
Then the instructor would interject with the first hint.
\begin{description}
\item[\underline{\textcolor{blue}{Hint 1}}] Draw a picture illustrating the problem, keeping in mind where the home is.
\end{description}
The general picture the students should have at this point could be as follows: 
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.7]{drone_schematic}
\caption{Schematic of the drone problem.}
\label{default}
\end{center}
\end{figure}

\begin{description}
\item[10 minutes] Discussion on possible ways to approach this problem.
\end{description}
\noindent The above discussion should include the method shown in \cite{mainreference}, in addition to a method (outlined in the next section) based on vectors. 
\begin{description}
\item[10 minutes] Have the students work (via their preferred method) on setting up a system of two equations.
\end{description}
\noindent Some flexibility is needed for the instructor here, since different groups may take on different approaches. For this reason, we have detailed the vector approach in the next section. Another hint will be injected by the instructor here:
\begin{description}
\item[\underline{\textcolor{blue}{Hint 2(a)}}] If the geometric approach is taken, can you identify two similar triangles, whose legs are components of the position and velocity vector of the drone, respectively? 
\item[\underline{\textcolor{blue}{Hint 2(b)}}] If the vector approach, can you identify the vector representation of the drone? 
\item[10 minutes] Let the students work out their various approaches until they obtain a pair of equations.
\end{description}
For the remaining time in class, briefly discuss how to derive a single equation from the pair of equations the students obtained in the last step. Even if the pair of equations are not entirely correct, the students will still have an idea of how to proceed to the next step once they correct their model equations. 

\begin{remark}
If an entire 50 minute class period is viewed to be too much time to devote to this problem, we suggest the following 20 minute trimmed down version of the above plan.
\begin{description}
\item[5 minutes] Have the students pair off and read the problem carefully, encouraging them to draw a picture.
\item[10 minutes] Discuss possible approaches to this problem (i.e. vectors vs. trigonometry).
\item[5 minutes] Have the students think for the remaining few minutes how to derive a pair of equations, or even a single equation directly!
\end{description}
The short in class version can be supplemented with the full beamer file, which the students can have access to if the instructor so wishes. 
\end{remark}


\section*{Solution with Vectors}
If we use the origin $(0,0)$ as home, then the initial position of the drone is $5$ miles east at $(5,0)$. 
Recall, the drone is heading home at a constant $b$ mph and the wind is blowing north at a constant $w$ mph.  Given the position curve, can you sketch the vector model at the point $(x,y)$?  You should draw the vectors: $\vec{p}$ (position vector), $\vec{w}$ (wind vector), $\vec{d}$ (drone vector), and $\vec{p}\,'$. (rate of change vector).

\begin{figure}[htbp]
\begin{center}
\includegraphics{DronePositionVector}
\caption{Drone Position Vector}
\label{DronePositionVector}
\end{center}
\end{figure}

Given any point $(x,y)$ on the position curve, the position vector is $\vec{p}=\langle x,y \rangle$.  Wind speed is a constant $w$ mph in the positive $y$ direction (north): $\vec{w}=\langle 0,w \rangle$. Thus, the rate of change of the position vector is given by $\vec{p}\,'= \vec{d} + \vec{w}$ where $\vec{d}$ represents the drone vector.  
Recall that the rate of change vector is tangent to the curve. See the Figure~\ref{DroneVectors} below.

\begin{figure}[htbp]
\begin{center}
\includegraphics{DroneVectors}
\caption{Drone vectors.}
\label{DroneVectors}
\end{center}
\end{figure}

Now we need to determine the equation of the drone vector.  Answer these questions.
\begin{enumerate}
\item What is the magnitude of the drone vector?
\item What is the direction of the drone vector?  
\end{enumerate}

\textbf{Answers}: 
\begin{enumerate}
\item $||\vec{d}||=b$ which represents the constant speed of the drone
\item It is in the opposite direction of the position vector.
\end{enumerate}

To find the equation of the drone position vector we need a unit vector in the opposite direction of the position vector and then multiply by the magnitude. Can you find the vector representation of $\vec{d}$?

\textbf{Answer}:
$$\vec{d}=-b\dfrac{\vec{p}}{||\vec{p}||}, \quad \text{NOTE}: ||\vec{d}||=b$$
So the vector equation becomes:
$$\vec{p}\,'= -b\dfrac{\vec{p}}{||\vec{p}||} +\vec{w}\, .$$
Breaking this equation into components and recalling the formula for the magnitude of a vector, we have:
$$\langle x', y' \rangle = -\dfrac{b \langle x,  y \rangle}{\sqrt{x^2 +y^2}} + \langle 0, w \rangle\, .$$
This gives us the system of equations: 
\begin{align*}
x' &= -\dfrac{bx}{\sqrt{x^2+y^2}} \\
y'&= -\dfrac{by}{\sqrt{x^2+y^2}}+w\, .
\end{align*}

\section*{Solving the main equation} \label{Sec:Analysis}
\subsection*{The main equations}
Regardless of the method taken by the student, eventually they should obtain the pair of equations
\begin{align} \label{xeqn}
\dfrac{dx}{dt}=\dfrac{-bx}{\sqrt{x^2+y^2}}\, ,
\end{align}

\begin{align} \label{yeqn}
\dfrac{dy}{dt}=\dfrac{-by}{\sqrt{x^2+y^2}} + w\, ,
\end{align}

\noindent which result in the following single differential equation for $y=y(x)$:
\begin{align} \label{singleeqn}
\dfrac{dy}{dx}=\dfrac{y}{x}-\dfrac{w}{b}\sqrt{1+ \left( \dfrac{y}{x}\right)^2}\, .
\end{align}
The analytical solution of (\ref{singleeqn}) is derived in \cite{mainreference}, and with initial condition $y(5)=0$, the following solution can be obtained via an appropriate substitution ($u=y/x$):
\begin{align} \label{mainsol}
y(x) =\dfrac{x}{2} \left[ \left( \dfrac{x}{5}\right)^{-w/b}-\left(\dfrac{x}{5}\right)^{w/b} \right], \quad x \neq 0\, .
\end{align}




\section*{Numerical Analysis}
\subsubsection*{Basic Euler Method}
Recall that for an initial value problem of the form
\begin{align} \label{IVP}
u'(x)=f(u,x), \quad u(x_0)=u_0
\end{align}
Euler's method is given by the numerical scheme
\begin{align} \label{Euler1}
u_{n+1} =u_n+ h f(u_n,x_n), \quad n=0,1,2,\ldots 
\end{align}
where $h>0$ is the step size. Now, (\ref{singleeqn}) can be written in the form of (\ref{IVP}), with 
$$f(y,x)=\dfrac{y}{x}-\kappa\sqrt{1+ \left( \dfrac{y}{x}\right)^2}$$
and where $\kappa \coloneqq w/b$. 
%however, for sake of clarity, we reproduce an Euler's method code below that can be used to output the exact solution of (\ref{singleeqn}) along with approximations. 

%\begin{remark}
The built in Mathematica Euler's method, via \texttt{ExplicitEuler}, can be used as in the included Mathematica notebook. One can adjust the value of $\kappa$, i.e.~the ratio of windspeeds and see the resulting trajectories. In particular, one will notice that for some values of $w$ and $b$, the drone will not deliver the package home (which intuitively should make sense). 
%\end{remark}
One could also employ the so-called Backward Euler method, which is given by the scheme
\begin{align} \label{Euler2}
u_{n+1}=u_n +h f(u_{n+1},x_{n+1})\, .
\end{align}
This method is also demonstrated in an associated Mathematica notebook. 


\subsubsection*{Improved Euler Method}
The scheme used for improved Euler's method (see e.g. \cite{LabODEbook}, Chapter 3) is 
\begin{align} \label{improvedEulerScheme}
u_{n+1}=u_n +\dfrac{h}{2} \left[ f(u_n, x_n) + f(u_n + h f(u_n, x_n), x_n +h)\right]\, .
\end{align}
For this scheme, Mathematica does not have a built in method, so we will need to write up a short script to use this method. See the included Mathematica notebook for this. 

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{ImprovedEulerOutput}
\caption{Output of the Improved Euler method for the drone problem with $\kappa=2/3$.}
\label{ImprovedEulerComparisonPlot}
\end{center}
\end{figure}

Figure~\ref{ImprovedEulerComparisonPlot} shows the output of this scheme with 10 steps. In this case $\kappa=2/3$ was used. 


\subsubsection*{Runge-Kutta Methods}
Usually, Runge-Kutta methods are not discussed in a first class on differential equations. However, this is certainly a topic that can be explored in depth for a student project. For this reason we have provided many details here, and a few relevant Mathematica notebooks. 

The first Runge-Kutta method is called the \textbf{midpoint method}, whose scheme is given by
\begin{align} \label{MidpointScheme}
\begin{split}
x_{n+1}&=x_n + h\\
u_{n+1}&=u_n + h f \left( x_n + \frac{h}{2}, u_n + \frac{h}{2} f(x_n, u_n) \right)\, .
\end{split}
\end{align}
This method converges at the same rate as the improved Euler method (that is, this scheme is  $O(h^2)$. This method can be implemented in Mathematica via \texttt{ExplicitMidpoint}; see the included Mathematica notebook. 

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{MidpointOutput}
\caption{Output of the Midpoint method for the drone problem with $\kappa=2/3$.}
\label{MidpointComparisonPlot}
\end{center}
\end{figure}

Figure~\ref{MidpointComparisonPlot} shows the output of this scheme with 10 steps and in this case $\kappa=2/3$ was used. 

An improvement of this method is a fourth order method, which is known as the \textbf{fourth order Runge-Kutta method}. We will abbreviate this by \textbf{RK4}. This scheme is given by:

\begin{align} \label{RK4scheme}
\begin{split}
x_{n+1}&=x_n+h\\
u_{n+1}&=u_n + \dfrac{1}{6} \left( k_1+2k_2 +2k_3+k_4 \right)\\
k_1&= h f(x_n, u_n), \\
k_2&= h f(x_n +h/2, u_n +k_1/2),\\
k_3&= h f(x_n +h/2, u_n + k_2/2),\\
k_4&= h f(x_n+h, u_n+k_3)\, .
\end{split}
\end{align}
This is quite a popular method in numerical analysis since it is relatively simple to implement (again, see the included Mathematica notebook!). The rate of convergence of this scheme is now $O(h^4)$. 

\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{RK4output}
\caption{Output of the RK4 method for the drone problem with $\kappa=2/3$.}
\label{RK4plotcomparison}
\end{center}
\end{figure}

Figure~\ref{RK4plotcomparison} shows the exact solution to the drone problem as well as the approximation obtained with the \textbf{RK4} method; in this case, 10 steps were taken, and the value of $\kappa=2/3$ was used. 
For more details and examples related to these numerical methods in Mathematica, see \cite{NumericalWebsite}. 


\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.7]{Comparison1}
\caption{Comparison of Numerical methods.}
\label{FirstComparisonPic}
\end{center}
\end{figure}



Now, a comparison of all of the methods discussed here with the exact solution can be done; see Figure~ \ref{FirstComparisonPic}. A closer look near the origin shows more clearly the differences in the methods; see Figure~\ref{FirstComparisonPicZoomed} below. 

\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.5]{Comparison2}
\caption{Comparison of Numerical methods.}
\label{FirstComparisonPicZoomed}
\end{center}
\end{figure}

The final comparison we can do is how long each computation takes; we do this via Mathematica's \texttt{Timing} command. The results can be found in Table~\ref{CPUtable}. 




Notice there is an order of magnitude difference between the $\textbf{RK4}$ method and the Improved Euler and Midpoint methods! This is the tradeoff for accuracy. Additionally, we see that the CPU time for the Improved Euler and Midpoint methods are approximately the same. 


\begin{table} [!t]
\centering
\begin{tabular}{l | c r}
Method       &       CPU Time (s)\\
\hline
Improved Euler & $8\times 10^{-6}$  \\
Midpoint        &    $8\times 10^{-6}$  \\
RK4       &   $0.00001$ \\
\end{tabular}
\caption{CPU timing for the drone problem.}
\label{CPUtable}

\vspace*{-.3 true cm}
\end{table}

\subsubsection*{Butcher Method}
A fifth order Runge-Kutta method was proposed by Butcher \cite{Butcher}: 
\begin{align} \label{ButcherRK5scheme}
\begin{split}
u_{n+1}&=u_n+ \dfrac{h}{90} \left( 7k_1+32k_3 +12k_4 +32k_5 +7k_6\right)\\
k_1&=f(x_n, u_n),\\
k_2&=f(x_n +h/4, u_n+k_1h/4),\\
k_3&=f(x_n+h/4, u_n+k_1h/8+k_2 h/8),\\
k_4&=f(x_n+h/2,u_n-k_2 h/2+k_3h),\\
k_5&=f(x_n+3h/4,u_n+3k_1h/16+9k_4h/16),\\
k_6&=f(x_n+h,u_n-3k_1h/7+2k_2h/7+12k_3h/7-12k_4h/7+8k_5h/7)\, .
\end{split}
\end{align}
An interesting student project would be to code up this method in Mathematica (or Matlab or other software) and compare with the previous methods. Be forewarned: the expected computation time for this method is \emph{much} longer than the previous methods! 











\begin{thebibliography}{99}

\bibitem{mainreference} 
Spindler, Richard. 2018. \textit{6-023-T-DroneHeadingHome}. 

\bibitem{LabODEbook}
Nagle, R.~Kent , Edward B.~Saff and Arthur D.~Snider. 2003. \textit{Fundamentals of Differential Equations, $6^{\rm th}$  Edition}. Reading MA:  Addison-Wesley. 

\bibitem{NumericalWebsite}
Dobrushkin, Vladimir. 2018. \textit{MATHEMATICA TUTORIAL for the First Course}. \href{http://www.cfm.brown.edu/people/dobrush/am33/Mathematica}{http://www.cfm.brown.edu/people/dobrush/am33/Mathematica}. 

\bibitem{Butcher}
 Butcher, J.~C. 1964.  On Runge-Kutta processes of high order.  {\it J. Australian Math. Soc.\/}. 4: 179--194. 

\end{thebibliography}
}

\newpage

\section*{Appendix: Calculus Review}
First some geometry review! Recall that the two triangles $ABC$ and $APQ$
\begin{center}
\begin{tikzpicture}[auto,scale=0.5,>=stealth]
 
\draw (0,0) -- (10,0) -- (10,6) -- (0,0);
\draw (7,0) -- (7,4.2);

\draw (-0.5,0) node {$A$};
\draw (7,-0.5) node {$P$};
\draw (10.5,0) node {$B$};
\draw (10,6.5) node {$C$};
\draw (7,4.9) node {$Q$};

\draw (3.5,-0.5) node {$\ell_1$};
\draw (8.5,-0.5) node {$\ell_2$};
\draw (10.5,3) node {$L_2$};
\draw (7.5,2.1) node {$L_1$};

\end{tikzpicture}\\
\end{center}
are similar if the ratio of their corresponding sides are equal (in the above picture, they are!). That is,
$$
\frac{L_1}{\ell_1} = \frac{L_2}{\ell_1+\ell_2}, \quad \frac{L_2}{L_1} = \frac{\ell_1+\ell_2}{\ell_1}\, .
$$


Recall that a \textbf{parametric curve} $\mathcal{C}$ is a two dimensional curve that can be described by the equations
$$(x,y)=(f(t), g(t))\,.$$
If $f,g$ are differentiable functions of $t$, and also $y$ is differentiable is a function of $x$, then the chain rule says that
$$\dfrac{dy}{dt} = \dfrac{dy}{dx} \cdot \dfrac{dx}{dt}\, ,$$
which means that
$$\dfrac{dy}{dx} = \dfrac{ \frac{dy}{dt}}{\frac{dx}{dt}}\, .$$
The value of $\dfrac{dy}{dx}$ gives the slope of the tangent to $\mathcal{C}$ at any point.
%\begin{example}
%The parametric curve given by $(x,y)=(t,t^2)$ describes which familiar object? \textbf{ANS}: parabola
%\end{example}
\begin{example}
What shape is the curve given parametrically by $(x,y)=(\sin^2 t, 2\cos(t)$)? HINT: How can you get to the familiar trigonometric identity $\sin^2(t) +\cos^2(t)=1$? \textbf{ANS}: A parabola opening to the left, given by $x=1-y^2/4$. 
\end{example}
\begin{example}
Does the parametric curve $(x,y)=(t^2-2t, t^3-3t)$ have a vertical tangent? HINT: consider $dy/dx...$ \; \textbf{ANS}: No; notice that $\frac{dy}{dx}=\frac{3t^2-3}{2t-2}$. A vertical tangent would occur when $2t-2=0$ and $3t^2-3\neq 0$; this does not happen. 
\end{example}
In $3D$, a parametric curve takes the form $\textbf{r}(t) = (x(t),y(t),z(t))$. A parametric surface in $3D$ is of the form $\textbf{r}(u,v)= (x(u,v), y(u,v), z(u,v))$. 
%\begin{example}
%Determine the surface in $3D$ given by $\textbf{r}(u,v)=(u, u\cos(v), u\sin(v))$. HINT: consider sums of squares of components... \textbf{ANS}: Cone opening along the $x$-axis given by $x^2=y^2+z^2$. 
%\end{example}
Recall that given a velocity vector $\textbf{v}$ in $2D$ or $3D$, the speed of an object with such velocity vector is given by the magnitude of this vector:
$$\text{speed}= || \textbf{v}|| = \sqrt{x^2+y^2+z^2}$$
if $\textbf{v}=(x,y,z)$. Now, if a particle's position is given by $\textbf{r}(t) = (x(t),y(t),z(t))$, its velocity is $\textbf{v}(t)=(x'(t),y'(t),z'(t))$. 



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