Astrodynamics Problems

There are several problem classes within scocp that are ready-made for common astrodynamics problems. These all require an integrator - in other words, by swapping the integrator, the same class can be used to design e.g. a rendez-vous trajectory in a restricted two-body dynamics, CR3BP, Clohessy-Wiltshire model, high-fidelity ephemeris model, etc.

Below is a list of implemented classes:

Continuous control problems

Class	Dynamics	Final time	Initial conditions	Final conditions
FixedTimeContinuousRdv	Translational dynamics	Fixed	Fixed	Fixed
FixedTimeContinuousRdvLogMass	Translational dynamics + log-mass	Fixed	Fixed	Fixed
FreeTimeContinuousRdv	Translational dynamics	Free	Fixed	Fixed
FreeTimeContinuousRdvLogMass	Translational dynamics + log-mass	Free	Fixed	Fixed
FreeTimeContinuousMovingTargetRdvLogMass	Translational dynamics + log-mass	Free	Fixed	Moving target
FreeTimeContinuousMovingTargetRdvMass	Translational dynamics + mass	Free	Fixed	Moving target

Log-mass vs. mass dynamics?

As the name suggests, there are classes that include a log-mass dynamics and mass dynamics, i.e. the state includes either \(z = \log{m}\) or \(m\). This makes a difference in the way control comes into the dynamics. With \(z\), the control \(\boldsymbol{u}\) is the acceleration, and we get

\[\begin{split}\dot{\boldsymbol{x}} = \begin{bmatrix} \dot{\boldsymbol{r}} \\ \dot{\boldsymbol{v}} \\ \dot{z} \end{bmatrix} = \begin{bmatrix} \boldsymbol{f}_{\mathrm{natural}} (\boldsymbol{x}(t), t) \\ 0 \end{bmatrix} + \begin{bmatrix} \boldsymbol{0}_{3\times3} & \boldsymbol{0}_{3\times1} \\ \boldsymbol{I}_3 & \boldsymbol{0}_{3\times1} \\ \boldsymbol{0}_{1\times3} & -1/c_{\mathrm{ex}} \end{bmatrix} \begin{bmatrix} \boldsymbol{u} \\ \| \boldsymbol{u} \|_2 \end{bmatrix}\end{split}\]

with the additional nonconvex constraint

\[0 \leq \| \boldsymbol{u} \|_2 \leq T_{\max} e^{-z}\]

In contrast, with \(m\), the control \(\boldsymbol{u}\) is the thrust throttle, and we get

\[\begin{split}\dot{\boldsymbol{x}} = \begin{bmatrix} \dot{\boldsymbol{r}} \\ \dot{\boldsymbol{v}} \\ \dot{m} \end{bmatrix} = \begin{bmatrix} \boldsymbol{f}_{\mathrm{natural}} (\boldsymbol{x}(t), t) \\ 0 \end{bmatrix} + \begin{bmatrix} \boldsymbol{0}_{3\times3} & \boldsymbol{0}_{3\times1} \\ \dfrac{T_{\max}}{m(t)} \boldsymbol{I}_3 & \boldsymbol{0}_{3\times1} \\ \boldsymbol{0}_{1\times3} & -\dfrac{T_{\max}}{I_{\mathrm{sp}}g_0} \end{bmatrix} \begin{bmatrix} \boldsymbol{u} \\ \| \boldsymbol{u} \|_2 \end{bmatrix}\end{split}\]

with the additional second-order cone constraint

\[ \| \boldsymbol{u} \|_2 \leq 1 \]

Here are some pros and cons to either model:

Property	Model with \(\log{m}\)	Model with \(z\)
Control type	Acceleration	Throttle
Affinity	Control-affine	General nonconvex
Additional constraint	Nonconvex	Second-order cone