# Continuous-Thrust Circular Orbit Phasing Optimization of Deep Space CubeSats
## 3 Spacecraft Dynamics and Trajectory Optimization  
Consider a two-dimensional, heliocentric mission scenario in which the MTC moves in the interplanetary space under the effect of the electric thruster-induced propulsive acceleration (\(\mathbf{a}\)) and the Sun's gravitational attraction. In this context, the orbital perturbations and uncertainty propagation [34] are neglected as is usually assumed in a preliminary phase of the trajectory analysis and design. Introduce a typical polar reference frame \(\mathcal{T}(O;r,\theta)\) whose origin coincides with the Sun's center of mass \(O\), in which the polar angle \(\theta\geq 0\) is measured counterclockwise from the Sun-spacecraft direction at the initial time \(t_{0}\triangleq 0\); see the scheme in Figure 5.  
Bearing in mind the expression of the propulsive acceleration vector \(\mathbf{a}\) given by Equation (16), the MTC equations of motion in \(\mathcal{T}\) are  
\[\frac{\mathrm{d}r}{\mathrm{d}t}=u,\quad\frac{\mathrm{d}\theta}{\mathrm{d}t}= \frac{v}{r},\quad\frac{\mathrm{d}u}{\mathrm{d}t}=-\frac{\mu_{\odot}}{r^{2}}+ \frac{v^{2}}{r}+\mathbf{a}\cdot\hat{\mathbf{r}},\quad\frac{\mathrm{d}v}{\mathrm{d}t}=- \frac{u\,v}{r}+\mathbf{a}\cdot\hat{\mathbf{\theta}} \tag{18}\]  
while the time variation in the spacecraft mass \(m\) is given by Equation (17) as a function of the solar distance \(r\) and the engine throttle dimensionless parameter \(\tau\). In Equation (18), \(u\) (or \(v\)) is the radial (or transverse) component of the MTC inertial velocity, \(\mu_{\odot}\) is the Sun's gravitational parameter, and \(\hat{\mathbf{r}}\) (or \(\hat{\mathbf{\theta}}\)) is the radial (or transverse) unit vector, which  
Figure 5: Heliocentric polar reference frame \(\mathcal{T}(O;r,\theta)\), in which the azimuthal angle \(\theta\) is measured from the Sun–spacecraft direction (green dash line) at the initial time instant. The unit vectors \(\hat{\mathbf{r}}\) and \(\hat{\mathbf{\theta}}\) indicate the radial and transverse directions, respectively, while \(\mathbf{a}\) is the MTC-induced propulsive acceleration vector. is in the positive direction of the Sun-spacecraft line (or the inertial velocity vector). The initial conditions, which are required to complete the differential system given by Equations (17) and (18), model the flight of the MTC along the initial parking orbit, whose plane contains at any time instant the thrust unit vector \(\mathbf{\hat{a}}\). In this regard, we assume a typical circular parking orbit of radius equal to \(r_{\odot}\), which approximates the Earth's heliocentric orbit with a zero value of the orbital eccentricity. Bearing in mind that the spacecraft initial mass is \(m_{0}=22.6\,\mathrm{kg}\), and assuming an initial polar angle equal to zero, the five (scalar) initial conditions are  
\[r(t_{0})=r_{\odot},\quad\theta(t_{0})=0,\quad u(t_{0})=0,\quad v(t_{0})=\sqrt{ \frac{\mu_{\odot}}{r_{\odot}}},\quad m(t_{0})=m_{0} \tag{19}\]  
The final conditions, i.e., the value of the spacecraft state variables at the end of the flight (time instant \(t=t_{f}>t_{0}\)), depend on the specific mission scenario to be analyzed. In this section, for example, we consider the typical case of a rapid circular orbit rising (or lowering), which also allows us to determine the optimal control law in terms of time variation in the pair \(\{\tau,\,\mathbf{\hat{a}}\}\). The optimal control law and the mathematical model used to optimize the spacecraft trajectory is employed (with minor changes) in the next section to analyze the MTC performance in a heliocentric phasing maneuver along a circular orbit. In the context of a rapid circle-to-circle orbit transfer, when the value of the final polar angle is left free and assuming a circular target orbit of assigned radius \(r_{f}\
eq r_{\odot}\), the three final (scalar) conditions related to the spacecraft state variables \(\{r,\,u,\,v\}\) are  
\[r(t_{f})=r_{f},\quad u(t_{f})=0,\quad v(t_{f})=\sqrt{\frac{\mu_{\odot}}{r_{f}}} \tag{20}\]  
in which \(t_{f}\) is an output of the optimization process.