

\subsection{Case Study Overview}
\label{subsec:case_overview}
To compare cooperative and conventional non-cooperative on-orbit refueling architectures,  simple multi-target refueling missions in LEOs are considered in this paper.
The target spacecraft are assumed to be deployed in multiple orbital planes with different inclination angles (i.e., $i=$53°, 53.2°, 70°, or 97.6° inspired by Starlink satellites).
We assume that
all of these orbits are circular with an altitude of 550 km, and the targets share the same Keplerian orbital elements except for the inclination and the argument of latitude. 
Table \ref{tab:parameters} lists the parameters used in this case study, and Fig.~\ref{fig:target_position} illustrates the initial positions of the targets and the servicer considered in the Earth-Centered Inertial (ECI) frame.
In this case study, the servicer separately refuels to Targets 1 to $n$ in Fig.~\ref{fig:target_position}.


\begin{table}[hbt!]
\center
\caption{Parameters employed for the case study}
\label{tab:parameters}
\begin{tabular}{lcr}
\thickhline
\multicolumn{2}{c}{Parameter}&Assumed value\\\hline
\multicolumn{3}{c}{\textit{Constellation Parameters}}\\
number of targets        &$n$     & 1 - 12\\
initial inclination [deg] &$i$& 53, 53.2, 70, 97.6\\
initial argument of latitude  &$u$&  See Fig.~\ref{fig:target_position} \\
altitude [km] && 550\\
\multicolumn{3}{c}{\textit{Refueling Mission Parameters}}\\
specific impulse of target [s] & $I_\mathrm{sp, t}$ & 300\\
specific impulse of servicer [s] & $I_\mathrm{sp, s}$ & 300\\
initial mass of each target [kg] & $m_\mathrm{t, I}$ & 1000\\
required refuel amount [kg] & $m_\mathrm{req}$ & 200\\\hline
\end{tabular}
\end{table}


\begin{figure}[hbt!]
\center
\includegraphics[width=.9\textwidth]{figs/target-eps-converted-to.pdf}
\caption{Initial positions of the targets and the servicer in the Earth-Centered Inertial (ECI) frame.}
\label{fig:target_position}
\end{figure}



To accomplish refueling to one target, either or both of the target and the servicer need to move to a rendezvous position. In this paper, we assume that these orbital transfers consist of two phases: inclination change and coplanar phasing.
For circular orbits, the required $\Delta v$ to change the inclination,\,$\Delta v_\mathrm{inc}$, is calculated as
\begin{equation}
\Delta v_\mathrm{inc} = 2 v \sin{\left( \frac{\Delta i}{2}\right)},
\label{eq:deltav_i}
\end{equation}
where $v$ and $\Delta i$ denote the velocity of the circular orbit $\sqrt{\mu/r}$ and the change in inclination, respectively.
The $\Delta v$ required for phasing maneuvers,\,$\Delta v_\mathrm{pha}$, is 
\begin{equation}
\Delta v_\mathrm{pha} = 2 \left | \sqrt{\frac{\mu}{r}} - \sqrt{\mu \left(\frac{2}{r} - \frac{1}{a}\right )} \right  |,
\label{eq:deltav_pha}
\end{equation}
where
\begin{align}
a &= \left( \frac{\alpha + 2 \pi k_2}{2 \pi k_1}\right) ^ {2/3} r,\\
\alpha &= 2\pi - \Delta u
\label{eq:deltav_pha_a}
\end{align}
$k_1$ and $k_2$ denote the number of complete revolutions during the phasing by the servicer and the target, respectively, and $\Delta u \in [0, 2\pi)$ denote the initial difference in the argument of latitude measured from the one that chases the other one. For the non-cooperative architecture, the servicer is always the one that chases the target, and in a cooperative architecture, it depends on their transfer strategies.
Please refer to Refs.~\cite{vall2013,sart2021} for the detailed derivation of Eq.~\eqref{eq:deltav_pha}.


This paper considers the following refueling architectures for the given LEO constellation:
\begin{enumerate}[label=\Alph*),ref=\Alph*]
\item \label{arc:trad}the conventional non-cooperative architecture where only the servicer actively conducts the orbital transfers;
\item a cooperative architecture where the inclination change is done by the targets and the phasing is done by the servicer;
\item a cooperative architecture where the inclination change is done by the servicer and the phasing is done by the targets;
\item \label{arc:extreme}an architecture where all targets come to the servicer while the servicer remains in the same orbit (i.e., fully cooperative targets and passive servicer ``depot''); and
\item \label{arc:optimum}the optimal cooperative case that minimizes the initial mass of the servicer.
\end{enumerate}
Figure \ref{fig:architecture} visually summarizes Architectures \ref{arc:trad}-\ref{arc:extreme}.
Architecture \ref{arc:optimum} can be any of Architectures \ref{arc:trad}-\ref{arc:extreme} or an architecture where both the servicer and the targets conduct both the inclination change and phasing maneuvers if that can reduce the initial mass of the servicer further. Note that the order of the targets to be refueled is the same among all architectures. The consideration of the optimum of this order is left for future work.

\begin{figure}[hbt!]
\center
\includegraphics[width=.99\textwidth]{figs/architecture.png}
\caption{Mission architectures of the multi-target refueling considered in this paper.}
\label{fig:architecture}
\end{figure}


As mentioned above, we consider both the inclination change and the coplanar phasing for each orbital transfer for refueling. The inclination $i$ and the relative argument of latitude $u$ to the initial argument of latitude of the servicer where the $j$\,th refuel is performed are denoted by $i_\mathrm{ref}(j)$ and $u_\mathrm{ref}(j)$, respectively, $i_\mathrm{init,t}(j)$ and $u_\mathrm{init,t}(j)$ represent the initial inclination and the initial relative argument of latitude of Target $j$, and $i_\mathrm{init,s}$ and $u_\mathrm{init,s}$ are those of the servicer. With this notation, each architecture has the following conditions:
\begin{enumerate}[label=\Alph*),ref=\Alph*]
\item $\forall j, i_\mathrm{ref}(j) = i_\mathrm{init,t}(j), u_\mathrm{ref}(j) = u_\mathrm{init,t}(j)$,
\item $\forall j, i_\mathrm{ref}(j) = i_\mathrm{init,s}, u_\mathrm{ref}(j) = u_\mathrm{init,t}(j)$,
\item $\forall j, i_\mathrm{ref}(j) = i_\mathrm{init,t}(j), u_\mathrm{ref}(j) = u_\mathrm{init,s}$, and
\item $\forall j, i_\mathrm{ref}(j) = i_\mathrm{init,s}, u_\mathrm{ref}(j) = u_\mathrm{init,s}$.
\end{enumerate}
For Architecture \ref{arc:optimum}, $i_\mathrm{ref}(j)$ and $u_\mathrm{ref}(j)$
are the design variables for the optimization, i.e., there are $2n$ design variables to be optimized. To find the optimal rendezvous points expressed by these parameters that minimize the initial mass of the servicer,\,$m_\mathrm{s,I}$, we use MATLAB's \texttt{MultiStart} solver. 


\subsection{Results}
\label{subsec:case_init_mass}

\subsubsection{Initial Mass of Servicer}
\input{case_res_init_mass.tex}

\subsubsection{Critical Mass Ratio}
\label{subsubsec:case_crit_mass_ratio}
\input{case_res_crit_mass_ratio.tex}



\subsection{Discussion and Sensitivity Analysis}
\label{subsec:case_sensitivity}
As previously mentioned in Sec.~\ref{subsec:case_init_mass}, Fig.~\ref{fig:mass_critical_case} suggests that the critical mass ratio generally increases as the number of targets increases in the same orbital plane; however, it does not necessarily increase when the newly added targets are in a different orbital plane. To study this behavior, in this subsection, the sensitivity of inclination angles on
the critical mass ratio is first examined by employing target sets where targets are distributed differently from Fig.~\ref{fig:target_position}. Table \ref{tab:target_sets} lists target sets considered in this analysis and their inclination angles. Note that each target set has the same set of the argument of latitude $u$ values as the original Target Set a (see Fig.~\ref{fig:target_position}).
These target sets are selected to explore the effects of multi-plane refueling compared to coplanar refueling (Target Set b and the other sets) and the large inclination change both near the initial orbit of the servicer (Target Sets c and d)  and far from that (Target Sets a and e).

\begin{table}[hbt!]
\center
\caption{Target sets and their inclination angles [$^{\circ}$].}
\label{tab:target_sets}
\begin{tabular}{lcccc}
\thickhline
Target \#                  & 1-3 & 4-6  & 7-9  & 10-12 \\\hline
a (original) & 53  & 53.2 & 70   & 97.6  \\
                             b            & 53  & 53   & 53   & 53    \\
                             c            & 53  & 53.2 & 53.2 & 53.2  \\
                             d            & 53  & 70   & 70   & 70    \\
                             e            & 53  & 53.2 & 70   & 75   \\\hline
\end{tabular}
\end{table}


Fig.~\ref{fig:mass_critical_case_diff_plane} illustrates the behavior of the critical mass ratio in response to the change in $n$ with different target sets listed in Table \ref{tab:target_sets}.
Since all targets in Target Set b are coplanar, the trend of this red line does not change even if all targets are in either $i=70^{\circ}$, $97.6^{\circ}$, or any other plane.
As expected, large drops of $\alpha_\mathrm{A-D}$ occur when a new Target $j$ is added to a new plane that is far from the plane where the Target $(j-1)$  is initially located. On the other hand, this result suggests that when the newly-added plane of Target $j$ is close to the plane of the Target $(j-1)$, the effect of adding a new plane becomes less significant. These behaviors can be observed by comparing the lines of Target Sets b, c, and d around $n=3$ and 4, and the lines of Target Sets a and e around $n=9$ and 10.
A comparison of the lines of Target Sets c and d also suggests that the increment of the critical mass ratio becomes larger when the difference of inclination between the initial plane and the newly added plane is larger.

Furthermore, for the coplanar case (Target Set b), $\alpha_\mathrm{A-D}$ for $n=11$ and 12 are almost the same value. This is because the newly added Target 12 is located close to the original position of the servicer indicating that this orbit is becoming crowded with targets and adding one more target affects the entire refueling architectures less significantly.
The line of Target Set c shows a similar trend: the slope of the Target Set c line becomes less steep around $n=12$. This is because target set c is almost coplanar whereas the orbital planes of Target Sets a, d, and e are more scattered.





\begin{figure}[hbt!]
\centering
\includegraphics[width=.7\textwidth]{figs/alpha_diff_plane-eps-converted-to.pdf}
\caption{Effect of different target sets on the critical mass ratio.}
\label{fig:mass_critical_case_diff_plane}
\end{figure}




We also compare the effect of the specific impulse of the target ($I_\mathrm{sp, t}$) and the servicer ($I_\mathrm{sp, s}$). Eq.~\eqref{eq:mass_critical_A-D} implies a smaller $I_\mathrm{sp, s}$ or a larger $I_\mathrm{sp, s}$ leads to a larger $\alpha_\mathrm{A-D}$, and conversely, a larger $I_\mathrm{sp, s}$ or a smaller $I_\mathrm{sp, s}$ leads to a smaller $\alpha_\mathrm{A-D}$.
Fig.~\ref{fig:mass_critical_case_diff_isp} shows the results of employing different values for both $I_\mathrm{sp, t}$ and $I_\mathrm{sp, s}$ for Target Set a (Fig.~\ref{fig:mass_critical_case_diff_isp_a}) and Target Set b (Fig.~\ref{fig:mass_critical_case_diff_isp_b}).
As can be seen from this figure, the sensitivity of each specific impulse turned out to be about the same level between both target sets.
Moreover, as expected from Eq.~\eqref{eq:mass_critical_A-D}, increasing $I_\mathrm{sp,t}$ or decreasing $I_\mathrm{sp,s}$ reduces the critical mass ratio implying the fully-cooperative case can be favored with smaller servicer mass.
On the other hand, smaller $I_\mathrm{sp,t}$ or larger $I_\mathrm{sp,s}$ makes the conventional non-cooperative architecture more favored.
The sensitivities of $I_\mathrm{sp, t}$ and $I_\mathrm{sp, s}$ also turn out to be the same according to this figure.
The investigation of the effect of employing electric propulsion is left for future work since the continuous orbital transfer significantly complicates the operation of this on-orbit refueling.



\begin{figure}[hbt!]
\begin{subfigure}{.49\textwidth}
\centering
\includegraphics[width=.99\textwidth]{figs/alpha_diff_isp_a-eps-converted-to.pdf}
\caption{Target Set a (multi-plane refueling).}
\label{fig:mass_critical_case_diff_isp_a}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\centering
\includegraphics[width=.99\textwidth]{figs/alpha_diff_isp_b-eps-converted-to.pdf}
\caption{Target Set b (coplanar refueling).}
\label{fig:mass_critical_case_diff_isp_b}
\end{subfigure}
\caption{Effect of different $I_\mathrm{sp}$ sets on the critical mass ratio.}
\label{fig:mass_critical_case_diff_isp}
\end{figure}








\subsection{Analytical Expression of Critical Mass Ratio}
\label{app:crit_mass_ratio}

From Eqs.\eqref{eq:mass_final_coop} and \eqref{eq:mass_final_noncoop}, the critical mass ratio can be derived as follows:
\begin{align}
    \begin{split}
        &{}m_\mathrm{s,F} \exp\left(\frac{\sum_{j = 1}^{n+1}\Delta v_\mathrm{s,c} (j)}{I_\mathrm{sp, s} g_0}\right) + \sum_{j = 1}^{n} \left\{ \left[\left( m_\mathrm{t,I} + m_\mathrm{req} \right) \exp\left(\frac{\Delta v_\mathrm{t, out} (j)}{I_\mathrm{sp, t} g_0}\right) - m_\mathrm{t,I}  \exp\left(\frac{-\Delta v_\mathrm{t, in} (j)}{I_\mathrm{sp, t} g_0}\right)\right] \exp\left(\frac{\sum_{k = 1}^{j}\Delta v_\mathrm{s,c} (k)}{I_\mathrm{sp, s} g_0}\right) \right\}\nonumber\\
        &{}=   m_\mathrm{s,F} \exp\left(\frac{\sum_{j = 1}^{n+1}\Delta v_\mathrm{s,n} (j)}{I_\mathrm{sp, s} g_0}\right) + \sum_{j = 1}^{n} \left(m_\mathrm{req} \exp\left(\frac{\sum_{k = 1}^{j}\Delta v_\mathrm{s,n} (k)}{I_\mathrm{sp, s} g_0}\right) \right)\nonumber
    \end{split}\\
    \begin{split}
   
        \Leftrightarrow &{}\frac{m_\mathrm{s,F}}{m_\mathrm{t,I}} \left(\exp\left(\frac{\sum_{j = 1}^{n+1}\Delta v_\mathrm{s,c} (j)}{I_\mathrm{sp, s} g_0}\right) - \exp\left(\frac{\sum_{j = 1}^{n+1}\Delta v_\mathrm{s,n} (j)}{I_\mathrm{sp, s} g_0}\right)\right)\\
        &{}= \sum_{j = 1}^{n} \left\{ \left( \frac{m_\mathrm{req}}{m_\mathrm{t,I}} \exp\left(\frac{\sum_{k = 1}^{j}\Delta v_\mathrm{s,n} (k)}{I_\mathrm{sp, s} g_0}\right) \right) -   \left[\left(1 +\frac{m_\mathrm{req}}{m_\mathrm{t,I}}  \right) \exp\left(\frac{\Delta v_\mathrm{t, out} (j)}{I_\mathrm{sp, t} g_0}\right) - \exp\left(\frac{-\Delta v_\mathrm{t, in} (j)}{I_\mathrm{sp, t} g_0}\right)\right] \exp\left(\frac{\sum_{k = 1}^{j}\Delta v_\mathrm{s,c} (k)}{I_\mathrm{sp, s} g_0}\right)\right\}\nonumber
    \end{split}\\
    \begin{split}
   
        \Leftrightarrow &{}\frac{m_\mathrm{s,F}}{m_\mathrm{t,I}} =  \frac{ \sum_{j = 1}^{n} \left\{ \left( \frac{m_\mathrm{req}}{m_\mathrm{t,I}} \exp\left(\frac{\sum_{k = 1}^{j}\Delta v_\mathrm{s,n} (k)}{I_\mathrm{sp, s} g_0}\right) \right) -   \left[\left(1 +\frac{m_\mathrm{req}}{m_\mathrm{t,I}}  \right) \exp\left(\frac{\Delta v_\mathrm{t, out} (j)}{I_\mathrm{sp, t} g_0}\right) - \exp\left(\frac{-\Delta v_\mathrm{t, in} (j)}{I_\mathrm{sp, t} g_0}\right)\right] \exp\left(\frac{\sum_{k = 1}^{j}\Delta v_\mathrm{s,c} (k)}{I_\mathrm{sp, s} g_0}\right)\right\}  }{\exp\left(\frac{\sum_{j = 1}^{n+1}\Delta v_\mathrm{s,c} (j)}{I_\mathrm{sp, s} g_0}\right) - \exp\left(\frac{\sum_{j = 1}^{n+1}\Delta v_\mathrm{s,n} (j)}{I_\mathrm{sp, s} g_0}\right)}.
    \end{split}
\end{align}



\subsection{Analytical Expression of Initial Mass of Servicer}













\section*{Nomenclature}


{\renewcommand\arraystretch{1.0}
\noindent\begin{longtable*}{@{}l @{\quad=\quad} l@{}}
$g_0$ & standard acceleration due to gravity\\
$I_\mathrm{sp}$ & specific impulse\\
$i$ & inclination\\
$m$ & general mass\\
$m_\mathrm{r}$ & total refuel mass\\
$m_\mathrm{req}$ & required refuel mass\\
$n$  & number of targets \\
$u$ & relative argument of latitude\\
$\alpha$ & critical mass ratio between a servicer and targets where the suggested mission architecture changes\\
$\Delta v$ & change in velocity\\
\multicolumn{2}{@{}l}{Subscripts}\\
c & cooperative case\\
F & mass after a whole campaign\\
f & mass after an orbital transfer\\
I & mass before a whole campaign\\
in & mass at inbound transfer\\
n & non-cooperative case\\
out & mass at outbound transfer\\
s & servicer\\
t & target\\
0 & mass before an orbital transfer
\end{longtable*}}




\section{Introduction}
\input{intro}


\section{Modeling Multi-Target On-Orbit Refueling Architectures}
\label{sec:problem_formulation}
\input{model}

\subsection{Comparison of Initial Mass of Servicer}
\label{sec:analytical_comparison}
\input{comparison}

\section{Case Study}
\label{sec:case}
\input{case_study}

\section{Conclusion}
\label{sec:conclusion}
\input{conclusion.tex}


\subsection{Overview of On-Orbit Refueling Architecture}
\label{subsec:coopoverview}
A general multi-target on-orbit refueling architecture considered in this paper follows the  
sequence below:
\begin{enumerate}[label=\arabic*),ref=\arabic*]
  \item a single servicer is launched and transferred to the orbital plane of the 1st target;
  \item \label{seq:2}the servicer rendezvous with the target, refuel it, and move to the next target; 
  \item repeat Phase~\ref{seq:2}) until all targets are refueled;
  \item \label{seq:3}the servicer goes back to its original position with the original inclination and the relative argument of latitude.
\end{enumerate}
In the case of a cooperative refueling architecture, both the servicer and a target are capable of 
orbital transfers to rendezvous in Phase \ref{seq:2} (Fig. \ref{fig:coop_seq}), 
whereas in the conventional non-cooperative architecture, only the servicer can conduct orbital transfers to the rendezvous point.
Note that cooperative target spacecraft consume some fuel to fly to a rendezvous point and come back to their original position after refueling (See Fig. \ref{fig:coop_seq}), and therefore, they need to receive more fuel from the servicer than those in the non-cooperative architecture to compensate this extra fuel consumption.




\begin{figure}[hbt!]
\center
\includegraphics[width=.9\textwidth]{figs/sequence.png}
\caption{Cooperative on-orbit refueling concept.}
\label{fig:coop_seq}
\end{figure}



\subsection{Initial Mass of Servicer}
\label{subsec:imserv}
To compare the cooperative and non-cooperative architectures introduced above, the mathematical expressions of the initial mass of the servicer in both architectures are derived in this subsection.
For the $j$\,th rendezvous (See Fig. \ref{fig:coop_seq}), the Tsiolkovsky rocket equation can be written as 
\begin{equation}
m_\mathrm{s,0}(j) =  m_\mathrm{s,f}(j) \exp\left(\frac{\Delta v_\mathrm{s} (j)}{I_\mathrm{sp, s} g_0}\right),
\label{eq:rocket}
\end{equation}
where $m_\mathrm{s,0}(j)$ and $m_\mathrm{s,f}(j)$ denote the mass of the servicer before and after the $j$\,th orbital transfer, respectively.
Since the initial servicer mass before the $j$\,th orbital transfer is the same as the servicer mass after the $\left(j-1\right)$\,th refuel, 
\begin{equation}
m_\mathrm{s,0}(j) =  m_\mathrm{s,f}(j-1) - m_\mathrm{r}(j-1),
\label{eq:mass_conservation}
\end{equation}
where $m_\mathrm{r}(j-1)$ represents the total refuel mass that the servicer gave to the $\left(j-1\right)$\,th target.
Modifying Eq.~\eqref{eq:rocket} to the $\left(j-1\right)$\,th orbital transfer and combining it with Eq.~\eqref{eq:mass_conservation} leads to 
\begin{equation}
m_\mathrm{s,0}(j-1) =  \left(m_\mathrm{s,0}(j) + m_\mathrm{r}(j-1)\right) \exp\left(\frac{\Delta v_\mathrm{s} (j - 1)}{I_\mathrm{sp, s} g_0}\right).
\label{eq:mass_j-1_to_j}
\end{equation}
From this equation, assuming the number of targets is $n$, the initial mass of the servicer at the beginning of the whole sequence, $m_\mathrm{s,0}(1)$, is derived as follows:
\begin{equation}
m_\mathrm{s,0}(1) =  m_\mathrm{s,0}(n) \exp\left(\frac{\sum_{j = 1}^{n-1}\Delta v_\mathrm{s} (j)}{I_\mathrm{sp, s} g_0}\right) + \sum_{j = 1}^{n-1} \left( m_\mathrm{r}(j) \exp\left(\frac{\sum_{k = 1}^{j}\Delta v_\mathrm{s} (k)}{I_\mathrm{sp, s} g_0}\right) \right).
\label{eq:mass_i_to_1}
\end{equation}
We further assume the final mass of the servicer is given (e.g., the dry mass) and treat Phase~\ref{seq:3}) as the $(n+1)$\,th orbital transfer. Thus, the given final mass is \,$m_\mathrm{s,F} = m_\mathrm{s,f}(n+1)$. From Eqs.~\eqref{eq:rocket} and \eqref{eq:mass_i_to_1}, the relationship between the initial mass,\,$m_\mathrm{s,I} = m_\mathrm{s,0}(1)$, and the final mass,  of the servicer of the whole scenario\,$m_\mathrm{s,F}$, is derived as follows:
\begin{align}
m_\mathrm{s,I} = m_\mathrm{s,F} \exp\left(\frac{\sum_{j = 1}^{n+1}\Delta v_\mathrm{s} (j)}{I_\mathrm{sp, s} g_0}\right) + \sum_{j = 1}^{n} \left( m_\mathrm{r}(j) \exp\left(\frac{\sum_{k = 1}^{j}\Delta v_\mathrm{s} (k)}{I_\mathrm{sp, s} g_0}\right) \right).
\label{eq:mass_final_to_1}
\end{align}

\subsection{Total Refuel Mass}
\label{subsec:mass_refuel}
For the cooperative architecture, a target comes from its original position to a rendezvous spot, and then the target goes back to the position where it would be if there was no active orbital transfer (see Fig. \ref{fig:coop_seq}). The rocket equations of both transfers by Target $j$ can be written as
\begin{align}
\label{eq:rocket_target}
m_\mathrm{t,in,f}(j) &=  m_\mathrm{t,in,0}(j)  \exp\left(\frac{-\Delta v_\mathrm{t, in} (j)}{I_\mathrm{sp, t} g_0}\right),\\
m_\mathrm{t,out,f}(j) &=  m_\mathrm{t,out,0}(j) \exp\left(\frac{-\Delta v_\mathrm{t, out} (j)}{I_\mathrm{sp, t} g_0}\right),
\end{align}
where $m_\mathrm{t,in,f}(j)$ and $m_\mathrm{t,in,0}(j)$ are the initial and the final mass of the first (inbound) transfer, $m_\mathrm{t,out,f}(j)$ and $m_\mathrm{t,in,0}(j)$ are those of the second (outbound) transfer (Fig. \ref{fig:coop_seq_mass}), and $\Delta v_\mathrm{t, in}$ and $\Delta v_\mathrm{t, out}$ are the $\Delta v$ required for each transfer. To fairly compare the cooperative and the non-cooperative architectures, the final mass of Target $j$ (after the second transfer),\,$m_\mathrm{t,out,f}(j)$, is assumed to be the same as the post-refuel mass of that Target in the non-cooperative architecture, i.e., a target in either architecture have the same amount of ``usable'' fuel supplied after every orbital transfer is completed.
Therefore, the relationship between the initial mass of the Target $j$, $m_\mathrm{t,in,0}(j)$, and the final mass of the Target $j$, $m_\mathrm{t,out,f}(j)$, becomes
\begin{equation}
m_\mathrm{t,out,f}(j) = m_\mathrm{t,in,0}(j) + m_\mathrm{req}(j),
\label{eq:mass_target_final}
\end{equation}
where $m_\mathrm{req}(j)$ is the required ``usable'' refuel amount by Target $j$ (i.e., the total refuel amount for the non-cooperative architecture). 
Furthermore, the relationship between the final mass after the inbound transfer and the initial mass before the outbound transfer is expressed as 
\begin{equation}
m_\mathrm{t,out,0}(j) = m_\mathrm{t,in,f}(j) + m_\mathrm{r,c}(j).
\label{eq:mass_t_refuel}
\end{equation}
where $m_\mathrm{r,c}(j)$ is $m_\mathrm{r}(j)$ for the cooperative architecture. From Eqs.~\eqref{eq:rocket_target} - \eqref{eq:mass_t_refuel}, the total refuel mass for Target $j$ is derived as follows:
\begin{equation}
\begin{split}
m_\mathrm{r}(j) &= \left( m_\mathrm{t,in,0}(j) + m_\mathrm{req}(j) \right) \exp\left(\frac{\Delta v_\mathrm{t, out} (j)}{I_\mathrm{sp, t} g_0}\right) - m_\mathrm{t,in,0}(j)  \exp\left(\frac{-\Delta v_\mathrm{t, in} (j)}{I_\mathrm{sp, t} g_0}\right).
\end{split}
\label{eq:mass_total_refuel}
\end{equation}
Substituting Eq.~\eqref{eq:mass_total_refuel} into Eq.~\eqref{eq:mass_final_to_1},
the initial mass of the servicer in the cooperative architecture, $m_\mathrm{s,I,c}$, can be analytically derived:
\begin{align}
\nonumber
m_\mathrm{s,I,c}& = m_\mathrm{s,F} \exp\left(\frac{\sum_{j = 1}^{n+1}\Delta v_\mathrm{s} (j)}{I_\mathrm{sp, s} g_0}\right)\\ 
&+ \sum_{j = 1}^{n} \left( \left[ \left( m_\mathrm{t,in,0}(j) + m_\mathrm{req}(j) \right) \exp\left(\frac{\Delta v_\mathrm{t, out} (j)}{I_\mathrm{sp, t} g_0}\right) - m_\mathrm{t,in,0}(j)  \exp\left(\frac{-\Delta v_\mathrm{t, in} (j)}{I_\mathrm{sp, t} g_0}\right)\right] \exp\left(\frac{\sum_{k = 1}^{j}\Delta v_\mathrm{s} (k)}{I_\mathrm{sp, s} g_0}\right) \right).
\label{eq:mass_final_coop}
\end{align}

\begin{figure}[bt!]
\center
\includegraphics[width=.7\textwidth]{figs/mass_sequence.png}
\caption{Change in mass during the $j$\,th cooperative refuel.}
\label{fig:coop_seq_mass}
\end{figure}




On the other hand, for the non-cooperative architecture, 
\begin{equation}
m_\mathrm{r,n}(j) =  m_\mathrm{req}(j).
\label{eq:mass_total_refuel_noncoop}
\end{equation}
where $m_\mathrm{r,n}(j)$ is $m_\mathrm{r}(j)$ for the non-cooperative architecture.
Therefore, substituting Eq.~\eqref{eq:mass_total_refuel_noncoop} into Eq.~\eqref{eq:mass_final_to_1},
the initial mass of the servicer in the conventional non-cooperative architecture, $m_\mathrm{s,I,n}$, can be analytically derived:
\begin{equation}
    m_\mathrm{s,I,n}= m_\mathrm{s,F} \exp\left(\frac{\sum_{j = 1}^{n+1}\Delta v_\mathrm{s,n} (j)}{I_\mathrm{sp, s} g_0}\right) + \sum_{j = 1}^{n} \left( m_\mathrm{req}(j) \exp\left(\frac{\sum_{k = 1}^{j}\Delta v_\mathrm{s,n} (k)}{I_\mathrm{sp, s} g_0}\right) \right).
\label{eq:mass_final_noncoop}
\end{equation}


