# Modeling and Analysis of Staged Constellation Deployment from a Single-Unit System
## 2 Defining the Deployment Problem
### The Deployment Problem  
The goal of the deployment strategy is to minimize the total phasing time, \(t_{phasing}\), which depends on the time spent in each deployment stage. Since, as shown in Equations (3) and (4), phasing times are governed by the initial and final relative motion, the objective is to determine suitable disengagement conditions at each stage that lead to an overall minimal phasing time, given by:  
\[t_{phasing}=\sum_{k=1}^{k_{max}}t_{k}\]  
Figure 3: Phasing of the final-stage satellite. Figure 2: Plot showing phasing branches. where \(t_{k}\) denotes the phasing time at stage \(k\). Rather than solving this as a formal optimization problem, an enumerative simulation approach is adopted. This involves selecting and assessing a discretized set of candidate values for the imparted \(\Delta v\) and the angles of separation at which disengagement into the next stage occurs and calculating the resulting phase times. The combination of parameters that yields the shortest total phasing time is then selected as the most effective deployment strategy. Since the final angular configuration of the constellation is a known mission parameter, the problem is formulated using a bottom-up approach. The final conditions at stage \(k_{max}\) are specified as \([\theta_{k_{max},\ i}\ 0]_{f}^{T}\), where \(\theta_{k_{max},\ i_{f}}\) is the desired final separation angle of the \(i^{th}\) satellite within the range \([0,\ 360)\). The initial conditions of Stage \(k_{max}\) are determined by the angle at which disengagement occurs in the previous stage, \(k=k_{max}-1\), and the change in the relative motion due to the disengagement \(\Delta v\). Whereby,  
\[\theta_{k_{max},\ i_{0}}=\theta_{(k_{max}-1),\ i_{f}}\ \text{and}\ \ \dot{\theta}_{k_{max},\ i_{0}}=\dot{\theta}_{(k_{max}-1),\ i_{f}}+\Delta \dot{\theta}_{DM}\]  
where \(\Delta\dot{\theta}_{DM}=f(\Delta v_{DM})\). Consequently, \(\left[\theta_{k_{max}-1,\ i}\ \dot{\theta}_{k_{max}-1,\ i}\ \right]_{f}^{T}\) represents the final relative motion of the penultimate stage. Once the initial conditions are found, Equations (3) and (4) can be employed to calculate \(\Delta t_{a}\) and \(\Delta t_{b}\) for the respective satellite. Similarly for stages \(k=2,\ \ldots,\ k_{max}\), the initial conditions of a satellite depend on the final relative motion of its parent satellite in the previous stage and the disengagement \(\Delta v\). For stage \(k=1\), the initial relative motion is determined solely by the disengagement \(\Delta v\), as the initial separation angle will always be zero; therefore, \(\theta_{1,10}=0\) and \(\dot{\theta}_{1,10}=\Delta\dot{\theta}_{DM}\). The final solution is, therefore, a cascaded variant of the one provided by [PERSON] et al. in [PERSON] et al. (2013), with the addition of the \(\Delta v\) imposed by the disengagement mechanism. Considering the case of an eight-satellite constellation requiring three deployment stages \((k=3)\), the phasing problem would require solving for the variables collated in Table 2.  
For each deployment stage, a reference-satellite pair will have unique values for \(\left[\theta\ \dot{\theta}\right]_{0}^{T}\), \(\left[\theta\ \dot{\theta}\right]_{f}^{T}\), and \(\Delta\dot{\theta}_{DM}\). Therefore, for a given \(\Delta v\) and desired \([\theta_{k_{max},\ i}\ 0]_{f}^{T}\), the values of \(\theta_{k,\ i_{f}}\) and \(\left(\Delta t_{a},\Delta t_{b}\right)_{k_{max}}\) that provide the optimal \(t_{phasing}\) need to be found. Extending the solution to all \(n\) satellites in the constellation yields a set of \(t_{phasing}\), the largest of which is the resulting total constellation phasing time.